For the homemade fettuccine: Combine the flour and salt in a stand mixer fitted with the dough hook. Add the eggs and yolks and mix on medium speed. Turn the mixer off and scrape down the sides of the bowl, pushing the flour towards the center. Continue to mix on medium speed, repeating these steps until the dough comes together into a ball. If the dough sticks to the sides of the bowl, add flour, 1 tablespoon at a time, until it forms a cohesive ball. Knead the dough on medium-high for 3 to 4 minutes. When the dough is smooth, bright yellow in color and bounces back when pressed, transfer it to a lightly floured work surface. Knead by hand a few more times and then form into a ball. The dough should feel moist but not sticky; knead in more flour, 1 tablespoon at a time, as necessary. Wrap the dough in plastic wrap and refrigerate for at least 2 hours but no more than 4 hours.
Fit the stand mixer with the pasta roller attachment and set to the widest setting (No. 1). Position the stand mixer with 2 to 3 feet of counter space to the left. Remove the dough from the refrigerator and divide into three portions. Work with one piece of dough at time, keeping the rest under plastic wrap so it doesn't dry out.
With your fingers, press a portion of the dough into a roughly 5-by-2 1/2-inch rectangle. Dust lightly with flour on both sides. Turn the mixer onto low speed and feed the dough through the roller with the 2 1/2-inch parallel to the roller. Next, fold the two ends of the dough into the center so they just meet and then fold one half over the other so the dough closes like a book; this is called laminating the dough. Use your fingers to seal the two edges together, then once again shape the dough into a 5-by-2 1/2-inch rectangle. Dust the dough with flour and feed it through the roller a second time. Repeat by laminating the dough one more time and feeding it through the roller. Dust with flour and feed through the roller three more times, dust with more flour as needed.
Adjust the roller to setting No. 3 and feed the dough through the roller twice. Adjust to setting No. 5 and feed the dough through two more times. At this point the dough will be over a foot long. Use your right hand to feed the dough through the roller and your left hand to catch the dough. Always be very gentle with the dough to prevent tearing. Be sure to feed the dough through the machine so that it does not fold onto itself. You can make small adjustments to how the dough is entering the roller by gently guiding it to the left or right. If the dough is too long to handle, simply cut it in half and work with a smaller portion.
Finally, adjust the roller to setting No. 7 and feed the dough through another two times. After the dough has gone through the roller for the last time, dust it all over with flour and then fold it over itself and hold under a piece of plastic wrap. Repeat these steps with the remaining two pieces of dough.
Replace the roller attachment with the fettucine cutter attachment. Dust a baking sheet with flour. Cut the rolled dough into 12-inch pieces and feed through the fettucine cutter on low speed. Twist the fettucine into a nest and place on the prepared baking sheet. Cover with plastic wrap while you cut the remaining dough. Lightly dust the tops of the fettucine nests with flour, cover with slightly dampened paper towel and then with plastic wrap. Hold at room temperature for no more than an hour or the pasta will dry out. (Makes 1 pound pasta dough.)
For the chicken and broccoli alfredo: Place a baking sheet in the oven and preheat the oven to 450 degrees F.
Mix the broccoli florets with 2 tablespoons of the olive oil, 1 teaspoon of the lemon zest, and salt and pepper to taste. Add the broccoli to the preheated baking sheet and roast until the lightly charred and tender, 18 minutes. Remove from the oven and let cool.
Bring a large pot of water to a boil. Season with salt so that the water tastes like the ocean.
Heat a 12-inch, high-sided pan over medium heat. Add the butter and the remaining 2 tablespoons olive oil. Sprinkle the chicken breasts with salt and pepper. When the butter is melted, add the chicken and cook until browned and cooked through, 4 to 5 minutes per side. If the chicken has browned on both sides but isn't cooked through yet, remove to a baking sheet and finish in the oven. Let cool and then cut into 1/2-inch pieces.
To the same pan, add the minced garlic; cook until fragrant, 1 minute. Add the flour and stir to combine, scraping up the dark bits from the bottom of the pan. Whisk in 1 cup of the half-and-half, making sure the flour is completely combined, without any lumps. Whisk in the remaining 3 cups half-and-half, bring to a boil and cook for 3 to 4 minutes. When the mixture has thickened around the edges, but before your whisk leaves a trail behind it, add the remaining 1 teaspoon lemon zest and the Asiago and Parmesan, and whisk to combine. Turn the heat to low and season with salt and pepper. The sauce should have the consistency of a thick soup. If the sauce thickens too much, add a little of the pasta water to thin it out. Add the chicken and broccoli to the cheese sauce and hold over low heat.
Meanwhile, cook to the pasta: Separate the strands of pasta on the baking sheet and add them to the boiling water. Cook for 30 seconds, or until the pasta floats to the top. Drain and add to the cheese sauce. Add the lemon juice and toss to combine. Garnish with parsley and serve immediately.
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