Halve the avocado, remove the pit, scoop the flesh out, and slice it thinly.
Spread the 2 tablespoons softened butter evenly on one side of each of the 4 bread slices.
Heat a nonstick or cast-iron skillet over medium-low heat.
Place two of the buttered bread slices in the skillet, butter-side down. Divide the 2–3 slices Monterey Jack evenly between the two bottom slices and layer them on the bread. Arrange the avocado slices over the Monterey Jack. Sprinkle salt and black pepper to taste over the avocado. Divide the 2–3 slices cheddar evenly and place them on top of the avocado.
Top each assembled sandwich with the remaining bread slices, butter-side up.
Cook the sandwiches over medium-low heat until the bottom bread is golden and the cheese begins to melt, about 3–4 minutes. Gently press with a spatula, flip each sandwich, and cook the other side until golden and the cheese is fully melted, about 3–4 more minutes.
Remove the sandwiches from the pan, let rest for about 1 minute, then slice and serve.