Cook the spaghetti in well-salted water until al dente. Save about 1 cup of pasta water before draining.
Season the chicken with Italian seasoning, oregano, basil, black pepper, and salt. Heat the olive oil in a large skillet and cook until golden-brown with seared edges and fully cooked through.
Add 1 tablespoon butter and half the garlic. Toss the chicken through the garlic butter for about 30 seconds, then transfer it to a plate.
Melt the remaining butter in the same skillet. Add the rest of the garlic, then stir in the heavy cream, milk, and cream cheese until smooth. Add the spinach and cook just until wilted.
Remove from the heat and stir in the mozzarella and Parmesan. Toss with the spaghetti until thick and glossy, using a little pasta water only if needed. Serve beside the chicken and finish with parsley.