Bring a large pot of salted water to a boil. Add the gnocchi and cook until they float, about 2–3 minutes, then drain and set aside.
In a large skillet over medium heat, melt 2 tablespoons of the butter. Add the cooked gnocchi in a single layer and cook without stirring until the bottoms are golden, about 2–3 minutes.
Use a metal spatula to flip the gnocchi and brown the other side until golden and crispy. Transfer browned gnocchi to a plate and season with salt and pepper.
Add the remaining 1 tablespoon butter to the skillet and melt over medium heat. Add the chopped zucchini and yellow squash and cook, stirring occasionally, until tender, about 3 minutes.
Stir in the minced garlic and shallot and cook for about 2 minutes until fragrant. Add the corn and grape tomatoes and cook for 1 minute to heat through.
Return the crispy gnocchi to the skillet. Stir in the Parmesan and chopped basil, taste and adjust with salt and pepper, then remove from heat.
Serve immediately, optionally garnished with extra Parmesan and basil.