Bring a large pot of lightly salted water to a boil. Cook linguine at a boil until tender yet firm to the bite, about 11 minutes; drain.
Whisk milk and cornstarch together in a bowl until smooth.
Melt butter in a saucepan over medium-low heat. Cook and stir minced garlic in hot butter until fragrant, 30 to 60 seconds. Whisk in milk mixture; season with salt and pepper. Bring mixture to a simmer, add Parmesan cheese, and whisk until cheese is completely melted and sauce is smooth. Stir in parsley and remove from heat.
Heat olive oil in a large skillet over medium-high heat. Saute sliced garlic in hot oil until fragrant, 30 to 60 seconds. Increase heat to high and stir zucchini, pinch of salt, and pinch of black pepper into skillet; cook and stir until zucchini is tender, about 5 minutes.
Stir sauce and Cheddar cheese into zucchini mixture; cook and stir until heated through, about 5 minutes more. Toss zucchini sauce and linguine together in a large bowl and garnish with basil.