Heat the 2 cups of vegetable oil in a small saucepan over medium-high to 350°F (175°C). Fry the 4 frozen mozzarella sticks for 2 to 3 minutes, turning once, until deep golden brown. Drain on a paper towel.

Lay out the 4 slices of bread. On 2 slices, layer 1 slice of American cheese, 2 fried mozzarella sticks, 1 more slice of American cheese, then close with the remaining bread.

Spread the 2 tbsp of softened butter on the outer faces of both sandwiches. Heat a skillet over medium for 1 minute, then cook the sandwiches for 3 to 4 minutes on the first side and 3 minutes on the second, pressing gently, until deep golden brown on both sides.

Transfer to a cutting board, slice diagonally, and serve with the 1 tbsp of marinara sauce on the side.