Preheat the oven to 500 degrees . Line a large baking sheet with foil. Scrub the sweet potatoes and cut into fries about 1/2 inch thick. Toss on the baking sheet with the olive oil and season lightly with salt. Spread in a single layer and bake for 10 minutes; flip with a spatula, then bake until the ends start to crisp, 8 to 10 minutes.
Heat a nonstick griddle or large skillet over medium heat. Lay the bread slices in pairs on a work surface. Spread the fig preserves equally onto 4 slices of bread. Lay 2 slices of cheese over the preserves. Arrange the prosciutto on top, folding evenly with the edges of the bread. Cover with the remaining bread slices and spread half the butter on top. Invert the sandwiches into the hot pan and spread the remaining butter on the tops. Cook, turning once or twice, until both sides are golden and crisp and the cheese is melted. Cut the sandwiches in half and serve with the sweet potato fries.