In a large pot of boiling salted water, cook pasta, stirring occasionally, until al dente according to package directions. Drain, then return to pot. Add Parmesan and fold until melted and smooth.
Meanwhile, in a large skillet over medium-high heat, melt butter. Once butter starts to bubble and turn lightly brown, add squash and zucchini and cook, stirring occasionally, until tender and lightly golden, about 10 minutes. Add garlic, stirring, until fragrant, about 1 minute more; season with 1 tsp. salt. Remove from heat.
Transfer pasta to a large serving bowl. Add zucchini mixture, basil, mint, and chives and toss to combine. Top with more Parmesan.
