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1
Melt butter in a large skillet over medium. Cook, swirling pan occasionally, until butter turns caramel-brown and smells nutty, 2-3 minutes. Add sage leaves and fry until crispy, 10-15 seconds. Remove from heat. Using a slotted spoon, transfer sage to paper towels to drain. Pour all but 2 Tbsp. brown butter into a small bowl.
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2
Cook panko and 4 sage leaves in same skillet, stirring to break up sage, until mixture is toasty, about 2 minutes. Transfer to a plate.
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3
Cook squash and 3 Tbsp. brown butter in same skillet over medium heat, stirring frequently, until squash begins to brown, about 5 minutes. Add 1/2 tsp. salt, 1/8 tsp. pepper, and 3/4 cup water; cover pan and cook until squash begins to soften, about 5 minutes. Remove lid and continue cooking, stirring occasionally, until liquid has evaporated and squash is tender and caramelized, about 5 minutes more.
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4
Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente, 8-10 minutes. Drain, reserving 3/4 cup pasta cooking liquid.
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5
Add remaining 3 Tbsp. brown butter to skillet and stir until squash is evenly coated. Add 1/2 cup pasta cooking liquid and simmer until a thin sauce forms, about 1 minute; season with 1 tsp. pepper. Add pasta, tossing to coat and adding pasta cooking water as needed to coat pasta. Remove from heat, top with half of the reserved sage, and stir to combine.
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6
Transfer pasta to a large serving bowl or individual pasta bowls. Top with panko mixture, remaining sage, and a generous shaving of Parmesan.
