Cut zucchini into sticks about 3" long and ⅛" thick.
Heat the vegetable oil in a skillet over medium heat. Fry the zucchini sticks in batches until they are a light-brown color. Transfer to a paper towel-lined plate to drain.
Boil the fusilli in salted water while you begin the sauce.
Dissolve the flour into the milk.
Melt half the butter and all of the olive oil in a new skillet. When the butter foams, turn the heat down to medium-low.
Add the flour and milk mixture, a little at a time, stirring constantly. When it has all been added, cook for about 30 seconds.
Add the fried zucchini strips and ¼ tsp salt, turning them 2-3 times in the sauce to coat them well.
Off the heat, add the remaining butter. Add in the egg yolk, then the cheese, mixing them in rapidly to keep the sauce creamy.
Taste and correct for salt.
Add the fusilli directly into the sauce and stir to coat each noodle. Add additional pasta water if the sauce is too thick.
Add in the basil and stir to incorporate completely. Serve immediately.