Cook the linguine in a large pot of well-salted water until al dente. Before draining, reserve about 1 cup of the pasta water.
Season the chicken with Italian seasoning, smoked paprika, garlic powder, black pepper, and salt. Heat the olive oil in a large skillet over medium-high heat and cook until golden-brown with lightly crisp seared edges and fully cooked through.
Add 1 tablespoon butter and half the minced garlic. Toss the chicken through the garlic butter for about 30 seconds, then transfer it to a plate and keep warm.
Melt the remaining butter in the skillet. Add the remaining garlic, heavy cream, milk, and cream cheese. Stir over low heat until smooth and slightly thickened.
Remove from heat and stir in the white cheddar, mozzarella, and Parmesan. Toss with linguine until every strand is heavily coated, adding pasta water only if needed. Serve beside the chicken and finish with parsley.