In large skillet over medium heat, heat oil. Add garlic; cook until golden and fragrant, about 3 minutes. Add kale; cook and stir until slightly wilted and dark green, 2-3 minutes. Season lightly with salt, to taste, and lemon juice. Remove to plate.
Wipe skillet clean. Spread butter on one side of each slice of bread, dividing evenly. Place two slices, buttered sides down, into skillet over medium-low heat. Onto each slice, divide half the Grana Padano; top with half the mozzarella.
Cook until cheese is melted and bottom is golden, 6-7 minutes. Add half the kale to one slice and close sandwich. Repeat with remaining ingredients.