Place vegetable oil in a Dutch oven or deep skillet and heat to 350°F. Slice zucchini into ⅛-inch rounds. Fry for 4-5 minutes or until lightly browned.
Drain the zucchini on a paper towel-lined plate and season with salt and pepper.
Bring a large pot of salted water to a boil and cook the spaghetti al dente according to package instructions. Save ½ cup of pasta water.
Add olive oil in large pan over medium heat and garlic, cooking for 1 minute. Add the zucchini, cooking for 2 minutes and break into small pieces with the edge of a wooden spoon. Add in the butter and basil, stirring until the butter melts.
Add the spaghetti to the sauce as well as ¼ cup of reserved pasta water and cheese, tossing to combine until creamy. Add more pasta water if needed. Serve with more grated cheese and fresh basil.
Notes
Work in batches when frying the zucchini so it gets cooked and doesn't lower the oil temperature.
Make sure the water you are cooking the pasta in is salted as some of the flavor comes from it.
DO NOT overcook the pasta when boiling. You want it just al dente or slightly under. It continues to cook with the sauce.
Don't use pre-shredded cheese. You want if finely grated so it melts well. Pre-shredded has added ingredients that will just make it clump up.
The butter can be omitted if you prefer. I like it as it adds some extra richness to the dish. You might need to add more pasta water to make up the difference.