In a small bowl, combine the scooped avocado, fresh lime juice, chopped cilantro, salt, and pepper. Mash with a fork until creamy but still slightly chunky. Set aside.

Lay out the four slices of sourdough bread. Spread an even layer of butter or mayonnaise on one side of each slice.

Flip two of the bread slices over (buttered side down). Divide half of the shredded cheese equally between these two slices.

Spread the mashed avocado mixture evenly over the cheese layer on both sandwiches.

Top the avocado with the remaining shredded cheese, and cover with the remaining bread slices (buttered side facing up).

Heat a large skillet over medium-low heat. Place the sandwiches in the skillet. Cover with a lid and cook for 4-5 minutes until the bottom is golden brown and the cheese begins to melt.

Carefully flip the sandwiches using a wide spatula. Cook uncovered for another 3-4 minutes until the second side is deeply golden and the cheese is completely melted.

Remove from the skillet, let rest for 1 minute, slice diagonally, and serve immediately.