For the ragu: To the bowl of a food processor, add the mushroom stems, celery, carrot, onion and garlic. Pulse until finely chopped but not pureed.
Heat a large Dutch oven with a couple tablespoons of olive oil over medium-high heat. Add the chopped veggies with a pinch of salt and crushed red pepper and cook, stirring occasionally, until soft and translucent, 5 to 7 minutes.
Add the sliced mushrooms and stir to combine. Cook for 2 minutes. Add the sausage and break up with a spoon, making sure to push down to the bottom of the pan so it can nicely brown. Brown the sausage for 5 minutes.
Add the wine and reduce by half, 2 to 3 minutes. Add the tomato paste and stir to combine. Add half of the chicken stock (2 cups), the thyme bundle, bay leaves and kosher salt to taste. Bring the mixture to a boil, then lower to a simmer and cook until the sauce has reduced by half, 10 minutes. Add the remaining chicken stock and cook until the sauce is thickened and holds its shape, 15 to 20 minutes more.
For the pasta: Mound the flour on a clean, dry work surface. Make a hole (this is also called a well) in the center of the flour pile that is about 8 inches wide (bigger is definitely better here). Crack all of the eggs and the yolk into the hole and add the olive oil, salt and water.
Using a fork beat the eggs together with the olive oil, water and salt. Then begin to incorporate the flour into the egg mixture with the fork; be careful not to break the sides of the well or the egg mixture will run all over your board and you will have a big mess! Also, don't worry about the lumps. When enough flour has been incorporated into the egg mixture that it will not run all over the place when the sides of the well are broken, begin to use your hands to really get everything well combined. If the mixture is tight and dry, wet your hands and begin kneading with wet hands. When the mixture has really come together to a homogeneous dough, THEN you can start kneading.
When kneading it is VERY important to put your body weight into it, get on top of the dough to really stretch it and not to tear the dough. Using the heels of your palms, roll the dough to create a very smooooooth, supple pasta dough. When done the dough should look VERY smooth and feel almost velvety. Kneading will usually take from 8 to 10 minutes for an experienced kneader and 10 to 15 for an inexperienced kneader. Put your body weight into it, you need to knead! This is where the perfect, toothsome texture of your pasta is formed. Get in there and have fun!
When the pasta has been kneaded to the perfect consistency, wrap it in plastic and let rest for at least 1 hour. If using immediately after 1 hour, do not refrigerate.
Divide the dough in half, then cut off a piece about the size of a tennis ball. (Keep the remaining dough wrapped.) Using your hands, flatten the dough and sprinkle with a little flour. Pass the dough through the pasta machine on the widest setting (#1), then fold each end towards the center like an envelope. Dust with more flour, then pass through the machine again. Repeat this process 3 to 4 times, folding the dough and flouring each time. Decrease the width to #2 and pass through the machine. Fold again and dust with flour. Continue to #3 and repeat, just folding and flouring once until you’ve reached #5.
Sprinkle a sheet tray with semolina flour and set aside. Trim the edges of the dough sheet to make a large rectangle. Divide it in half by cutting down the center, then cut the dough into roughly 2-by-1 1/2-inch rectangles. Roll each rectangle around a wooden garganelli dowel to create a cylinder with a triangular tip on either end, and use a little water to secure the seam. Set onto the sheet tray with semolina flour and repeat the rolling, cutting and shaping process with the remaining fresh pasta dough.
To serve: Set up a large pot of boiling water and generously season with kosher salt. It should be as salty as the sea. Add the garganelli and cook for 2 to 3 minutes. (Work in batches if you like.)
Meanwhile, add the desired amount of sauce to a saute pan and place over medium heat to warm. Add the cooked garganelli directly from boiling water into the ragu along with a ladle of pasta water. Toss to combine until all the noodles are nicely coated. Add the parmesan cheese and a big drizzle of extra-virgin olive oil. Toss to combine. Plate, then top with more parmesan and a dollop of fresh ricotta cheese.