Heat the oven to 400°F. Trim the stems and roots from the beets so the flesh is exposed and they can stand upright. Place them cut-side down in a small deep baking dish with ¼ inch of water covering the bottom. Cover tightly with foil, prick a few holes in the top, and roast for 45 to 60 minutes until very tender. Check halfway through and add more water if needed.

When the beets are cool enough to handle but still warm, peel off the skins with your hands or paper towel. Roughly chop and transfer to a small blender, then purée until completely smooth. Add water one tablespoon at a time if needed to help blend.

For each pasta dough (you'll make two separate batches), mix the beet purée, eggs, and yolks in a bowl until well combined. Add flour to a food processor, then pour in the liquid mixture. Pulse until the mixture resembles coarse crumbs, then turn out onto your work surface and knead by hand until smooth and elastic, about 8 to 10 minutes. Wrap each dough separately and let rest at room temperature for at least 30 minutes and up to 2 hours.

While the dough rests, toast your walnuts in a dry skillet over medium heat for 3 to 4 minutes until fragrant and lightly golden. Set aside to cool, then coarsely chop.

Pulse the ricotta, goat cheese, and Parmigiano-Reggiano in a food processor until completely smooth. Season generously with salt and pepper, then add chives and lemon zest. Pulse briefly just until combined. Transfer to a piping bag or bowl and refrigerate until ready to use.

Roll out your rested dough to the appropriate thickness for ravioli, typically setting 6 or 7 on a pasta machine. Use your ravioli mold to form the hearts, piping about 1 teaspoon of filling into each cavity. Cover with a second sheet of pasta, seal well, and use a rolling pin to press firmly. Cut into decorative rounds using a 1½-inch fluted cookie cutter.

Bring a large pot of water to a gentle boil and season generously with salt. In a large skillet over medium-high heat, add ¼ cup pasta cooking water and bring to a lively simmer. Reduce heat to low and whisk in cold butter one piece at a time until melted and emulsified. The key here is keeping the heat low and whisking constantly to prevent the sauce from breaking. Season with salt and keep warm on the lowest heat.

Gently drop the ravioletti into boiling water, shaking off excess flour. Stir briefly to prevent sticking. Cook for 2½ to 3 minutes, stirring occasionally, until tender. Always taste-test one to check doneness.

Transfer the cooked ravioletti to the butter sauce using a spider sieve or slotted spoon, then swirl the pan until well coated. Divide among plates and spoon extra butter sauce on top. Finish with toasted walnuts and a generous sprinkle of fresh herbs. Serve immediately while hot.