Bring a pot of heavily salted water (at least 1 tablespoon of kosher salt) to a boil and cook spaghetti in it according to directions until one minute BEFORE al dente.
Meanwhile, heat the olive oil (2 tablespoons) over medium high heat in a large rimmed skillet or pot (a dutch oven works great).
Sauté the onion in the olive oil over medium high heat until softened and starting to brown, about 2 minutes.
Add the garlic to the onions, stir and sauté until fragrant (about 30 seconds).
Add the zucchini, summer squash, and asparagus to the onions and garlic, along with a pinch of salt. Stir to coat and sauté until vegetables start to soften and asparagus begins to turn a bright green, 2-3 minutes.
Add the cherry tomatoes to the rest of the vegetables and sauté for 1-2 minutes, until they start to heat up.
Just before the pasta is al dente, transfer it directly from the pot to the skillet (using tongs or a slotted spoon). Use a ladle to scoop about 1 cup of the pasta water to the skillet as well. (alternatively, you can reserve some pasta water, drain it in a colander, and add it to the skillet from there).
Add the grated parmesan cheese and stir together, allowing the cheese to melt into the liquid in the skillet.
Add the baby spinach a handful at a time, folding the pasta over it so it wilts down.
Tear up half of the burrata (it should come in two, four-ounce balls, so use only one for now) and add it to the skillet, stirring together until melted and the cream seeps into the sauce. Season to taste with kosher salt and black pepper
Add the fresh basil and stir together.
By now, most of the pasta water should have been absorbed by the spaghetti or evaporated, leaving behind a creamy, thick sauce. If this hasn't happened yet, continue heating, stirring occasionally, until thickened.
Just before serving, tear the remaining burrata on top of the pasta. If you like, you can drizzle it with more olive oil and/or some aged balsamic vinegar or balsamic glaze, and sprinkle it with more basil and parmesan cheese.