Preheat a non-stick skillet. Spread out the walnuts and toast, tossing occasionally, until golden, about 3 minutes. Let cool, then coarsely chop; set aside.
Cook the pasta in a large pot of boiling salted water, according to package directions, stirring occasionally, until al dente. Drain, reserving 1 cup of the pasta cooking liquid.
Meanwhile, heat the oil in a large skillet over medium-high heat. Add the squash and cook, tossing occasionally, until crisp-tender, about 5 minutes. Add the garlic and 1/8 teaspoon red pepper flakes and cook, stirring, just until garlic begins to brown, about 2 minutes. Immediately add ½ cup pasta cooking liquid to keep garlic from burning; reduce heat to low and gradually add butter, swirling skillet and adding more pasta cooking liquid as needed, until a thick, glossy sauce forms.
Add the pasta to the skillet with squash and sauce and toss to coat. Add the lemon juice, ¼ cup Parmesan, 2 tablespoons mint, and remaining 1/8 teaspoon red pepper flakes and toss to combine; season with salt and pepper.
Serve pasta topped with reserved walnuts, more Parmesan, and some mint.
DO AHEAD: Walnuts can be toasted a few days ahead of time. Store airtight in the refrigerator.