Roasting the beets brings out a natural sweetness and makes the colors vibrant. Place trimmed beets cut-side down in a baking dish with a bit of water. Cover with foil and roast at 400°F for 45–60 minutes until tender. Once slightly cooled, peel and purée until smooth.
Combine beet purée, eggs, and yolks in a bowl or food processor. Mix until fully incorporated, then knead into a smooth dough. Cover and let rest for 30 minutes to 2 hours to relax the gluten for easier rolling.
Pulse ricotta, goat cheese, and Parmigiano-Reggiano in a food processor until smooth. Add chives and lemon zest, mixing just until combined. Optionally, squeeze in a little lemon juice for brightness. Keep the filling in a piping bag or bowl in the fridge until ready to assemble.
Roll out the dough thinly and cut into desired shapes using a fluted cookie cutter or ravioli mold. Pipe or spoon a small amount of filling onto each piece and fold over to seal. Ensure the edges are tight to prevent filling from leaking during cooking.
Boil a large pot of salted water and gently drop in the ravioletti. Cook 2½–3 minutes until tender. Meanwhile, melt butter in a skillet over low heat, whisking in pasta cooking water to create a smooth sauce. Transfer cooked pasta to the pan and coat in the butter.
Plate the ravioletti and spoon extra butter sauce on top. Sprinkle toasted walnuts and fresh herbs over the pasta for a visually stunning finish. Serve immediately for best texture and flavor.