Step 1

Cook pasta in a large pot of generously salted boiling water until al dente. Reserve 1 1/2 cups water; drain pasta.

Step 2

Heat a large skillet over medium-high heat. Swirl in oil. Add anchovies and garlic; cook, stirring, just until anchovies break down and garlic turns golden, about 30 seconds. Add zucchini; season with salt and pepper flakes. Cook, stirring occasionally, until zucchini is very tender, about 5 minutes. Add pasta and 1 cup reserved pasta water. Simmer, stirring occasionally, until liquid is reduced and slightly thickened, 1 to 2 minutes.

Step 3

Remove from heat; stir in cheese and mint. Add more pasta water if needed, a few tablespoons at a time, until sauce evenly coats pasta. Serve, drizzled with oil and sprinkled with more mint, pepper flakes, and cheese.