Dust a baking sheet with semolina. Divide the pasta dough into 4 balls. Using a pasta machine, roll each ball out to a thin sheet, progressing gradually down from the widest to the thinnest setting. Lay each sheet on a lightly floured work surface. Cut the pasta dough into 2-inch squares.
Lay a thin dowel or a pencil diagonally across a bottom corner of each square, and roll up to form a quill. Slip the garganello off the dowel onto the prepared baking sheet. Repeat with all the pasta squares. Cover with a damp towel.
In a heavy-duty 6- to 8-quart pot, heat the olive oil over high heat until smoking. Add the onion, carrot, and celery and cook until softened and lightly brown, 8 to 10 minutes. Add the wine, Basic Tomato Sauce, anchovy paste, red pepper flakes, and rosemary and bring to a boil.
Season the meat with salt and pepper and add to the tomato sauce. Return to a boil, reduce the heat to a simmer, and cook until the meat falls apart with the poke of a fork, about 1 1/2 hours. Remove from the heat and let cool for 10 minutes.
Transfer the stew, about 1/2 cup at a time, to a food processor, and pulse just until it has the consistency of a thick sauce. Pour into a saucepan, taste, and adjust the seasoning if needed.
Bring 6 quarts of water to a boil in a large pot and add 2 tablespoons salt. Meanwhile, bring the ragu to a boil. Drop the garganelli into the boiling water and cook until tender, 1 to 2 minutes. Drain.
Add the pasta to the ragu and toss gently to coat. Transfer to a warmed serving bowl and serve immediately.
Mound the flour in the center of a large wooden board. Make a well in the center of the flour and add the eggs. Using a fork, beat together the eggs and then begin to incorporate the flour, starting with the inner rim of the well. As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. (Don't worry if it looks messy; when half of the flour is incorporated the dough will begin to come together.) Start kneading the dough, using primarily the palms of your hands. Once the dough is a cohesive mass, remove the dough from the board and scrape up and discard any leftover bits.
Lightly re-flour the board and continue kneading for 10 minutes, dusting the board with additional flour as necessary. The dough should be elastic and a little sticky. Wrap the dough in plastic and let rest at room temperature for 30 minutes before using.
Basic Tomato Sauce
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until softened and light golden brown, 8 to 10 minutes. Add the thyme and carrot and cook until the carrot is quite soft, about 5 minutes. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat, and simmer until it's as thick as hot cereal, about 30 minutes. Season with salt to taste. This sauce can be refrigerated for up to 1 week or frozen for 6 months.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.