Bring a large pot of lightly salted water to a boil. Cook linguine at a boil until tender yet firm to the bite, 8 to 10 minutes; drain.

Heat olive oil in a large skillet over medium-high heat. Add garlic, cook and stir until golden brown. Add squid; cook until it turns white. Stir in white wine and cherry peppers; bring to a simmer and cook until wine is reduced by half, about 3 minutes.

Stir cornstarch into cream in a small bowl, then add to simmering calamari mixture; season with basil, pepper flakes, salt, and black pepper. Stir until sauce is thickened.

Add linguine to sauce; toss until evenly coated. Sprinkle with Parmesan cheese.