Bring a large pot of water to a boil for the pasta.
In a large skillet, heat 2 tbsp. EVOO, 2 turns of the pan, over medium heat. Add the zucchini and lightly brown, 6 to 8 minutes. Add the onions and half of the garlic; season with salt and pepper. Reduce heat; keep warm until the pasta is done.
Meanwhile, in a food processor, pulse the mint, parsley, remaining garlic, nuts, cheeses, lemon zest and juice, and about 1/4 cup EVOO to form a pesto; season to taste. Transfer the pesto to a large bowl.
Salt the boiling pasta water and cook the tagliatelle until al dente. Stir 1 cup hot cooking water into the pesto. Drain the pasta and add to the bowl; toss to combine, then gently toss with the zucchini and serve.