Heat oven to 400 degrees F. Trim stems and roots from beets, place cut-side down in a small deep baking dish with 1/4 inch of water, cover tightly with foil, prick a few holes, and roast 45 to 60 minutes until very tender. Check halfway through and add water if dish looks dry.

Cool beets until handleable, then peel skins off by hand or with a paper towel. Roughly chop and blend until smooth, adding water 1 tablespoon at a time only if needed. Cool completely before using in dough.

For each dough, mix beet puree, eggs, and yolks until combined. Add flour and knead by hand about 8 to 10 minutes until smooth and elastic, or pulse in food processor until crumbly then knead by hand. Wrap tightly and rest at room temperature at least 30 minutes.

Pulse ricotta, goat cheese, and Parmigiano-Reggiano in food processor until smooth. Season with salt and pepper. Add chives and lemon zest and pulse briefly. Add a small squeeze of lemon juice if desired. Refrigerate until ready to use.

Roll each rested dough to setting 6 or 7 on a pasta machine or very thin by hand. Use ravioletti mold to form pasta, piping small amounts of filling into each cavity without overfilling. Cover with second dough sheet and press firmly with a rolling pin to seal. Cut each ravioletto with a 1.5-inch fluted cookie cutter.

Bring a large pot of well-salted water to a gentle boil. In a large skillet over medium-high heat, add 1/4 cup pasta cooking water and bring to a simmer. Reduce to low and whisk in cold butter one piece at a time until emulsified and silky. Season with salt and keep warm on lowest heat.

Shake excess flour from ravioletti and gently drop into boiling water. Stir briefly to prevent sticking. Cook 2 1/2 to 3 minutes until tender, tasting one piece to check. Transfer with spider sieve to butter sauce and swirl to coat. Serve immediately with toasted walnuts and fresh herbs.