Combine the flour, beet root powder, xanthan gum, and salt in a large mixing bowl. Make a well in the center and add 1 tablespoon olive oil and 4 eggs. Scramble the eggs slightly and start mixing them into the flour mixture. Continue to mix until it forms a dough which is slightly sticky. If the dough is dry and crumbly, you can add another egg.
Turn the dough onto a lightly floured surface and knead for 3 – 4 minutes. If the dough is too sticky, knead in a little more flour. Wrap in plastic wrap and let sit for 10 minutes. Unwrap the dough, knead once more for 2 – 3 minutes, then wrap in plastic wrap and let rest for 30 minutes.
Divide the dough into four pieces. Keep the dough you are not using at the time wrapped to prevent the dough from drying out. Roll one piece of dough out with a rolling pin until it is thin enough to run through the pasta machine. Dust with a little flour, run through the machine at the widest setting, fold in half, and run through again. Continue to do this until the dough is smooth, 3 or 4 times. Adjust the machine, one setting down at a time, and continue to do so until the desired thickness is achieved. On our machine, 3 was perfect. Run the pieces of dough through the fettuccine cutting blade and form in nests to sit until cooking.
Bring a large pot of salted water to a boil. Add the pasta and cook for 3- 4 minutes. Save about 1 cup of pasta cooking water and drain.
Combine walnuts, garlic, wine, cream, ¼ cup olive oil, ¼ cup Parmesan, 1 teaspoon salt, and pepper in a blender and blend until smooth and creamy. If too thick add a little water to thin it out.
Return the pot the pasta was cooked onto to the stove and add about ½ cup of the reserved pasta water. Add the sauce and cook over medium heat for about 2 minutes. Add the cooked pasta and stir gently to coat. If the sauce is too thick, add some reserved pasta water.
Serve with parsley, a drizzle of olive oil, more cheese, and some fresh pepper.