Place the zucchini slices in a colander and sprinkle with salt. Let them sit for about 15 minutes to draw out excess moisture. Rinse and pat dry with paper towels.
In a skillet over medium heat, add olive oil. Once hot, add the zucchini slices and sauté for about 5-7 minutes until they are tender and slightly golden. Remove from heat and set aside.
Preheat a separate skillet or griddle over medium heat. Butter one side of each slice of bread.
On the unbuttered side of four slices, layer the cheddar cheese, sautéed zucchini, mozzarella cheese, garlic powder, and black pepper. Top with the remaining bread slices, buttered side up.
Grill the sandwiches for about 3-4 minutes on each side, or until the bread is golden brown and the cheese is melted.
Remove from the skillet, slice in half, and serve warm.