Preheat a countertop griddle to 350F or place a large skillet over medium heat. Brush lightly with avocado oil.
Slice the zucchini on the diagonal into 16 slices, about 1/2 inch thick each, and lay on the griddle or in the pan (for a skillet, you may need to work in batches). Cook until lightly browned on both sides, about 3 minutes per side.
Transfer to a paper towel lined plate and let sit to drain a few minutes. Pat the tops dry with a rag or more paper towel.
In a small bowl, whisk together the salt, pepper, garlic powder, and paprika. Sprinkle one side of each zucchini slice with the seasoning mixture.
Pair the zucchini slices by approximate size, and divide the sliced cheese between them, to make 8 sandwiches. Brush the griddle or pan with a little more oil or some butter and lay the sandwiches down on the grill. Let cook until the cheese is melted.