Heat 1 tablespoon olive oil in a large skillet over medium-high heat.
Add the diced chicken breast. Season with Italian seasoning, paprika, salt, and black pepper.
Cook the chicken for 6 to 8 minutes, stirring occasionally, until browned and fully cooked. Check that the thickest pieces reach 165°F / 74°C.
Reduce the heat to medium-low. Add 1 tablespoon butter and the minced garlic. Cook for 30 seconds, stirring constantly, until fragrant.
Add the cream cheese and milk. Stir until the cream cheese melts into a creamy garlic sauce that coats the chicken.
Remove the skillet from the heat. Stir in ½ cup mozzarella and ½ cup cheddar cheese until melted. Reserve the remaining cheese for assembling the wraps.
Place a tortilla on a clean work surface. Add about ¼ cup mozzarella and cheddar cheese in the center, then spoon one-quarter of the creamy garlic chicken filling over the cheese.
Fold the sides of the tortilla inward, then roll it tightly into a wrap. Repeat with the remaining tortillas, chicken filling, and cheese.
Lightly grease a clean skillet with butter or olive oil and heat it over medium heat.
Add the wraps seam-side down. Cook for 2 to 3 minutes per side, pressing gently with a spatula, until golden brown, crisp, and the cheese is melted.
Let the wraps rest for 1 to 2 minutes. Slice in half and serve with parsley, red pepper flakes, ranch, garlic aioli, or hot sauce if desired.