Preheat oven to 400F.

Whisk 2 eggs, egg yolks, 2 tablespoons water and 1 tablespoon oil in a medium bowl. Add all-purpose flour, whole-wheat flour and 1/4 teaspoon salt to the bowl of a stand mixer fitted with a dough hook. With the mixer on medium speed, add the egg mixture and knead until the dough is smooth and elastic, about 5 minutes. (Add more all-purpose flour, 1 tablespoon at a time, if the dough is too sticky to come together into a ball.) Place the dough in an oiled bowl, cover and let rest at room temperature for at least 30 minutes or up to 2 hours.

Meanwhile, combine squash, onion, garlic, pepper, 2 tablespoons oil and 1/2 teaspoon salt on a rimmed baking sheet. Bake, stirring occasionally, until the squash is golden brown and tender, about 25 minutes. Transfer the vegetables to a food processor, add Parmesan and puree until smooth.

Dust 2 rimmed baking sheets with flour. Put a large pot of water on to boil.

Set up a pasta machine and turn it to the largest setting. Cut the dough into 4 pieces. Working with one piece at a time, press it into a rectangle. Run the dough through the machine four times, folding it in half on itself each time before running through the machine again .

Turn the pasta machine setting one notch smaller and run the dough through once. Continue turning the machine to smaller settings and running the dough through until it is about 1/16 inch thick. Place the sheet of pasta on a lightly floured work surface. Repeat with the remaining pieces of dough.

Whisk the remaining egg and 1 tablespoons water together in a small bowl. Determine approximately where the halfway point is lengthwise on a sheet of pasta. Brush one half of the dough with egg wash. Using about 2 teaspoons each, spoon mounds of the filling onto the egg-washed side, leaving about 3/4 inch between the mounds. Fold the other half of the pasta sheet over to cover the filling. Seal the edges around each mound of filling and press out any excess air. Use a pastry cutter or knife to cut individual ravioli. Transfer the ravioli to the prepared baking sheets. Repeat with the remaining pasta sheets and filling.

Add half the ravioli to the boiling water and stir a few times to submerge and separate them. Cook at a gentle boil until just tender, 2 to 3 minutes. Using a slotted spoon, transfer the ravioli to a serving bowl. Repeat with the remaining ravioli.

Heat butter and the remaining 2 tablespoons oil in a medium skillet over medium heat. Cook, swirling the pan to encourage even browning, until the mixture starts to turn brown and smell nutty, about 2 minutes. Season with the remaining pinch of salt. Pour the mixture over the ravioli and gently toss to coat.