Preheat the oven to 300 degrees F or preheat grill over high heat.
Toss the peppers with some oil, place on a baking sheet, and roast in the oven until blackened. Turn them over, blacken on the second side, and remove to cool. Alternatively, blacken on a grill in the same manner. Gently remove the skins and seeds; do not rinse with water. Place in a food processor bowl and pulse to puree. Scrape down sides and pulse a few more times.
Add to the processor bowl 1 egg, 2 1/2 teaspoons olive oil, and salt; pulse to combine. Add in 12 ounces of the semolina and pulse to combine. A dough ball should start to form after 6 to 7 pulses. If the mixture is too wet, add more semolina until a dough ball forms. Process for 1 minute and turn the dough out onto a lightly semolina-floured cutting board. Sprinkle with a little more semolina and knead for 2 to 3 minutes. Wrap tightly in plastic wrap and allow to rest at room temperature for 30 to 45 minutes.
Prepare a work area for rolling out the pasta: a large cutting board, all-purpose bench flour, large sheets of plastic wrap, dough blade or knife, small amount of water, and a pastry brush.
Cut a 1/2-inch thick piece of the dough from the ball and shape into an oval, making sure that the dough isn't wider than the width of the pasta roller. Process through the pasta machine by folding the dough into thirds, feed through the rollers on level 1, and repeat until the dough is smooth, elastic, and in an even shape. Add more flour or a sprinkle of water, if necessary, to adjust the consistency of the dough. Continue rolling out through level 9, without folding, and cutting the sheets into 12 to 14-inch long pieces as needed. Wrap carefully and individually in plastic wrap; it's okay to layer. Repeat until the dough is used up.
Combine the cheeses with the pepper and whisk in 2 eggs. On a clean, lightly semolina-floured surface lay out a sheet of pasta. With a round tablespoon measure, lightly press some cheese mixture into the measuring spoon and "thonk" with your finger gently onto the sheet, spacing them 1 1/2 inches apart. Lightly brush beaten eggs around the cheese. If the dough is wide enough you can fold it over the cheese, or if it's thinner in width, center the cheese spacing and add a top sheet. Start at the center and gently compress around the cheese to push out any air. Cut between the raviolis with a bench scraper, knife, or pizza cutter. Dust raviolis lightly with semolina flour and set on a cookie rack; repeat until all the raviolis have been made. Reserve the extra cheese for garnish.
Bring a large pot of water to boil over high heat and add 1 tablespoon kosher salt. When ready, add the raviolis gently into boiling water, pressing into the water as needed, and cook for 2 to 3 minutes. Remove from the water, drain, and add to the Chardonnay Cream Sauce. Place into a serving dish and drizzle some of the Basil Oil. Sprinkle with a bit of the remaining cheese, the minced parsley, and serve immediately.
In a small saucepan over medium-high heat, add the butter and when foamy, add the shallots and cook for 5 to 6 minutes. Add the chardonnay and the salt and pepper and bring to a boil. Allow to simmer for 10 minutes. Add the cream and combine well. Simmer, stirring occasionally, for 10 to 15 minutes; the mixture will thicken slightly. Hold warm until service. Adjust seasonings as needed.;
Prepare an ice bath of ice and water. Bring a large pot of water to a boil over high heat, add the basil leaves, and cook for 10 seconds. Remove immediately to the ice bath. When cool, transfer to a salad spinner or strainer and remove water.
Add the leaves and oil into a blender and pulse to puree. Scrape down the sides and pulse a few more times. Strain the mixture through 4 layers of rinsed cheesecloth into a sterile canning jar (16-ounce size is good). This process will take about an hour.
Refrigerate unused portion for up to 1 week.