PREP: This dish goes quickly once it's started, so shred the zucchini and summer squash ahead of time. Grate both on the large holes of a cheese grater. Dice the onion, mince the garlic, and we're ready to start!
PASTA: Bring a large pot of water to the boil with 1 tablespoon of fine sea salt. Cook the spaghetti according to package directions, subtracting 2 minutes (the pasta will finish cooking later). Drain the pasta and set aside (don't run it under cold water).
ONION AND GARLIC: While the pasta cooks, melt the butter in a very large skillet or large pot over medium high heat (we'll add the pasta into this pan later, so use a big one). Once butter is mostly melted, add in the onion and garlic. Cook, stirring frequently, until the onion is becoming golden, about 3-4 minutes.
ZUCCHINI AND SQUASH: Add the shredded zucchini and squash. Stir over medium-high heat for 1 minute. Add seasonings: 1/2 teaspoon of each -- salt, pepper, and paprika. Continue to cook, stirring frequently, until most of the liquid is absorbed and veggies are tender.
CREAM AND CHEESE: Add in the chicken stock, heavy cream, freshly grated Parmesan cheese, and frozen corn. Stir and bring to a simmer. Reduce the heat and simmer until thickened a bit, about 2-3 minutes.
PASTA: Add in the hot drained pasta and toss vigorously for 2-3 minutes, over medium heat, with tongs or two kitchen spoons. Continue to toss until the pasta is well coated and the sauce thickens. Once pasta is well coated, add in the pesto and toss again to combine. Taste and adjust, adding more salt and pepper, if needed. Add an extra splash of heavy cream, chicken stock, or pesto to thin the sauce a bit if desired.
SERVE: Serve immediately garnished with fresh herbs and add additional parmesan to each plate.