Melt about 2 tablespoons of the unsalted butter in a large, non-stick skillet over medium heat until the foaming subsides. Reduce the stovetop heat to medium-low and add a handful of fresh sage leaves or so, swirling to toss the sage in butter. Add 2 bread slices and cook, swirling occasionally, until pale golden brown on the bottom side, about 2 minutes. Flip the bread and toast the other side for another 2 minutes or so.
After the second side has been toasted, transfer to a cutting board. Place 1 ounce of Tillamook cheddar cheese on top of each slice. Use a spoon to dollop a tablespoon or so of apple-cranberry chutney on top of the cheese, spreading it evenly across the cheese. Close the sandwich.
Melt another tablespoon of butter in the skillet over medium-low heat. Add the sandwich and cook, swirling occasionally and using a flexible metal spatula to occasionally press down on the sandwich, until one side is a deep, even golden brown, about 5 minutes. Add the remaining butter and flip the sandwich. Cook until this side is a deep golden brown and serve immediately with the sage leaves, which at this point are crisp from all that butter and gentle frying. Serve immediately, and repeat with any remaining ingredients to make additional sandwiches.