Cook the spaghetti in a large pot of salted boiling water until al dente. Reserve ½ cup pasta water, then drain.
Season the small chicken pieces with smoked paprika, garlic powder, black pepper, Italian seasoning, and salt.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook for 5–7 minutes until golden-brown with lightly caramelized edges and 165°F inside.
Add 1 tablespoon butter and half the minced garlic. Toss the chicken briefly until fragrant and glossy with garlic butter, then transfer it to a plate.
Melt the remaining butter in the same skillet and cook the remaining garlic for about 30 seconds. Lower the heat and pour in the heavy cream and milk. Add the cubed Velveeta and stir gently until completely melted into a smooth golden cheese sauce.
Remove from the heat and gradually stir in ¾ cup cheddar and the mozzarella until melted. Add the spaghetti and gently toss until the sauce coats every strand. Add reserved pasta water only if necessary, keeping the cheese sauce thick and abundant.
Scatter the remaining cheddar over the hot spaghetti and allow it to melt. Arrange the small garlic butter chicken pieces beside a wide, loosely spread serving of cheesy spaghetti. Finish with parsley and spoon extra sauce around the pasta.