For the pasta: You can prepare the dough either using a stand mixer or by hand.
Using a stand mixer: In the bowl of a stand mixer fitted with the hook attachment, add the flours and mix to combine. Add the eggs and mix on medium speed until the dough come together and forms a ball. If the dough is dry, add a little water, 1 teaspoon at a time. Reduce the speed to low and mix until the dough is smooth, about 6 minutes. Wrap the dough with plastic wrap or a damp towel. Let it sit at room temperature for about 30 minutes.
By hand: On a wooden table or in a bowl, create a volcano shape with the flours. Then make a large hole in the middle. Break the eggs inside of the volcano. Whip the eggs with a fork to create a uniform color (until you can't see the difference between egg yolk and egg white). Then, little by little with the tips of your fork, start to incorporate the flour, starting from the center and moving out (never break or pass beyond the volcano). If the dough is dry, add a little water, 1 teaspoon at a time. When the mixture is uniform, use your hands and knead the dough until smooth. Wrap the dough with plastic wrap or a damp towel. Let it sit for about 30 minutes.
Flour the table and roll out the dough until very thin (until you can just see your hands through the dough). For pappardelle, roll the dough up into a log, lightly flouring the dough as you roll it. Slice into about 1-inch-thick slices. Unroll the slices and lay out the strips flat onto a floured sheet tray until ready to cook.
For the sauce: When ready to cook the pasta, bring a pot of water to a boil and add salt. Cook the pasta for 3 minutes.
Meanwhile, in a pan, bring the vegetable stock to a simmer. Add the butter and cooked pasta. Season it with freshly milled black pepper and salt. At the very end, add the Parmigiano-Reggiano little by little, stirring it in as you go. Garnish with fresh herbs and more cheese.