Special equipment: a pasta machine
For the pasta: Combine the flour and 2 teaspoons salt on the countertop and make a well in the center. Add the eggs and yolks to the well and beat with a fork. Slowly work the flour into the eggs until the dough comes together. If it feels a little sticky, add a little more flour, about 1 teaspoon at a time, until it is just no longer sticky. Wrap tightly in plastic wrap. Let rest at room temperature for 30 minutes.
For the meat filling: Meanwhile, heat the oil in a medium skillet over medium heat. Add the carrot, onion and garlic and cook, stirring occasionally, until soft, about 8 minutes. Increase the heat to medium-high and add the sausage and 1/2 teaspoon salt. Cook, stirring and breaking up any clumps, until browned, about 4 minutes. Add the tomato paste and cook, stirring, 1 minute. Add a splash of water to loosen the mixture ever so slightly.
For the ricotta-sage filling: Stir together the ricotta, parmesan, sage, zest, egg, 1/2 teaspoon salt and some pepper in a bowl.
For the tomato sauce: Heat the oil in a medium saucepan over medium heat. Add the garlic and cook, stirring, until fragrant, about 2 minutes. Add the crushed red pepper flakes and stir 30 seconds. Add the tomatoes, granulated garlic, onion powder, thyme, a pinch of salt and 1/2 cup water and bring to a simmer. Simmer until slightly thickened, about 10 minutes. Rewarm before serving.
Cut the pasta dough into 6 equal pieces. Work with one piece at a time, leaving the remaining pieces covered in plastic wrap until ready to roll out. Flatten the dough into an approximately 1/4-inch-thick rectangle and dust it with flour. Roll the dough out on the widest setting of your pasta roller. Fold the dough into thirds and roll it through that setting again. Continue rolling out the dough, dusting with additional flour as necessary, adjusting the pasta roller to a narrower setting each time you roll, until the sheet of dough is a little thicker than 1/16th of an inch (usually the second to last setting on your roller).
Lightly flour a sheet pan and set aside. Dust your work surface with a little flour and lay the pasta sheet down. Spoon a scant tablespoon of one of the fillings (alternate or mix and match) along the bottom part of the dough, leaving about an inch of space between each spoonful. Next, lightly brush the bottom edge and the spaces in between the filling with water. Fold the pasta in half to cover the filling and press the dough down around the filling, starting from the closed seam and working your way forward between each ravioli, pushing out as much air as possible, before sealing the front edge of the pasta. Use a knife or a pastry cutter to cut the individual ravioli. Press the sides with the tines of a fork to create indents along the border. Transfer the ravioli to the floured sheet pan and dust with a little additional flour. Roll out the remaining dough in the same manner and fill. Bring a large, wide pot of salted water to a boil. Add the ravioli and cook, adjusting the heat as necessary to maintain a gentle simmer, until the pasta is just tender (if you cut into one, you will no longer see a raw white line inside), about 4 minutes. Drain gently. Divide the ravioli among 6 shallow bowls and top with tomato sauce. Serve with additional Parmesan and garnish with the basil leaves.