Gruyère, cut into 1/4 inch- or 1/8 inch-thick slices
fresh basil leaves
Add the olive oil to a large skillet set over medium-high heat. Add the zucchini coins in an even layer on the bottom of the pan. Sprinkle the zucchini with salt and let them fry in the oil until deep golden, about 3 to 5 minutes. Flip and fry the other side the same way as the first.
Transfer the zucchini to a plate and set aside.
In a small bowl, combine the soft butter and Parmesan, mixing together with a rubber spatula until well combined.
Slather the Parmesan-butter on one side of each piece of bread. (Leftover butter can be wrapped and refrigerated for up to 1 week.)
On the unbuttered side of two pieces of bread, add a couple pieces of Gruyère followed by half of the zucchini coins. Top with fresh basil and a couple more pieces of Gruyère. Add the two remaining pieces of bread to the tops of the sandwiches, butter side up.
Sear the sandwich in a skillet set over medium-low heat. Cook until the cheese is melted and the bread is deep golden.
Remove from the pan, slice in half, and serve warm.