Preheat the skillet over medium heat.
Spread 1 tablespoon of softened butter on one side of each slice of sourdough bread.
On the unbuttered side of two slices, layer half of the shredded white cheddar cheese.
Evenly distribute the peach slices on top of the cheese.
Drizzle 1 tablespoon of honey over the peach slices and sprinkle a pinch of salt if desired.
Top with the remaining white cheddar cheese, and close the sandwiches with the other slices of bread, buttered side up.
Place the sandwiches in the skillet and cook for about 3-5 minutes, or until the bread is golden brown and the cheese begins to melt.
Carefully flip the sandwiches using a spatula and cook for another 3-5 minutes on the other side until golden brown and the cheese is fully melted.
Remove from heat and let cool for a moment before slicing in half.
Serve immediately, drizzled with additional honey if desired.