Roast the squash: Heat the oven to 400°F (200°C). Halve the butternut squash lengthwise, scoop out the seeds, and place cut-side down on a baking sheet. Roast until very tender and lightly caramelized, about 45 minutes. Cool until easy to handle, then scoop out the flesh and mash or purée until smooth.
Make the filling: In a bowl, combine 2 cups mashed roasted squash, ricotta, 1/2 cup grated Parmesan, thyme, nutmeg, salt, and pepper. Mix until smooth. Chill the filling while you make the dough (it’s easier to portion when cool).
Make the pasta dough: Mound the flour on a clean work surface and make a wide well in the center. Add eggs, olive oil, and a pinch of salt. Use a fork to whisk the eggs, gradually pulling flour into the center until a shaggy dough forms.
Knead and rest: Knead the dough 8–10 minutes until smooth and elastic (add a tiny bit of flour if sticky). Cover with a damp towel or wrap and rest 30 minutes at room temperature.
Roll the pasta: Divide dough into 2–4 pieces. Keep unused pieces covered. Roll with a pasta machine (or rolling pin) into thin sheets—you should almost see your hand through the dough. Cut sheets into long strips about 4 inches (10 cm) wide.
Fill and seal: Spoon small mounds of filling (about 1 teaspoon to 1 tablespoon, depending on size) onto one strip, spacing them about 2 inches apart. Lightly brush water around each mound. Lay a second strip on top and press around the filling to remove air and seal.
Cut ravioli: Use a ravioli cutter, stamp, or sharp knife to cut individual ravioli. Press edges firmly (use a fork if needed). Transfer to a flour-dusted baking sheet and keep covered with a towel.
Cook: Bring a large pot of well-salted water to a gentle boil. Add ravioli and cook 3–4 minutes, or until they float and the pasta is tender. Remove with a slotted spoon or spider.
Make the brown butter sauce: While the ravioli cook, melt butter in a skillet over medium heat. Add sage leaves and cook until the butter turns golden-brown and smells nutty and the sage is crisp. Stir in toasted walnuts.
Toss and serve: Add drained ravioli to the skillet and gently toss to coat (add a splash of pasta water if needed). Serve immediately with Parmesan on top.