Melt the butter in a large skillet over medium heat. Carefully swirl the butter around in the skillet. Cook until the butter starts to turn brown and the white foam clears. It will begin to have a sweet caramel-like nutty smell; about 2 to 3 minutes.
Drop the sage leaves into the butter scattering them around so they are spaced evenly around the pan. (Use caution as the fresh leaves will spatter as they cook.) Fry the sage until the leaves are crisp; about 15 to 20 seconds. Remove the skillet from the heat.
Use a pair of tongs or a slotted spoon t0 transfer the sage to the plate with paper towels to drain.
Pour the brown butter, into a bowl, leaving about 2 tablespoons in the skillet. Set aside.
Toast the breadcrumbs
Add the bread crumbs to the same skillet. Crumble up 4 or 5 of the cooled sage leaves and add them into the skillet. Cook the bread crumbs until they are toasted; about 2 minutes, stirring constantly. Transfer the breadcrumbs and sage to a plate.
Cook the Squash & the Pasta
Add the squash and about 3 tablespoons of the reserved brown butter to the same skillet over medium heat. Cook stirring often until squash starts to brown; about 5 minutes. This will give the squash a nutty buttery taste and bring out the natural sweetness of the squash.
Season with salt and freshly cracked black pepper, to taste. Pour in ½ cup water, then cover the skillet and cook for 3 to 5 minutes or until the squash softens a little. Then remove the lid. Continue to cook stirring constantly until the liquid evaporates and the squash is tender and caramelized; about 3 to 5 minutes more.
Meanwhile, cook pasta in a large pot of boiling heavily salted water, per package instructions. Reserve 1 cup of the pasta water before draining!
Toss the cooked squash with the remaining brown butter until evenly coated. Then add about ¼ cup pasta water. Simmer for about a minute until a thin sauce forms. Toss the pasta in the skillet with the squash. Add more pasta water, as needed, to coat the pasta. The pasta water helps loosen up the noodles and keeps them from sticking together. Remove the skillet from the heat. Toss with half of the sage leaves.
Divide the pasta among serving dishes. Top with bread crumbs, sage, and as much shaved Parmigiano-Reggiano as you desire.
What to Look For and When To Find Delicata:
Delicata squash is available from late summer through early winter and if stored properly can be kept from 3 to 6 months in a cool dark place.
Choose squash with smooth unblemished skin feel firm, especially near the stem ends.
Choose squash that are similar in shape and size, so that they cook evenly and consistently.
Substitute Delicata with any of the following vegetables:
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