Bring a large pot of salted water to a boil. Add fettucini and cook al dente (according to package instructions). Reserve 1 cup of pasta water and drain the rest. Set noodles aside while you prepare the asiago sauce.
In the original skillet, melt butter over medium heat. Add garlic and cook for 1 minute. Whisk in the flour, Italian seasoning, salt, and red pepper flakes. Cook until golden brown, about 1 minute.
Slowly drizzle in heavy cream, whisking constantly. Add in asiago, spinach, and ¼ cup of the reserved pasta water. Add more pasta water as needed to reach your desired consistency. Cook until the spinach is wilted, stirring frequently for 1-2 minutes.
Add cooked fettucini, tossing to coat.
Slice chicken into strips. Top the pasta with the sliced chicken.
Garnish with asiago cheese and parsley. Serve immediately.