Combine the cherries, wine, shallots, sugar, vinegar and salt in a small saucepan over medium heat. Bring to a simmer and cook, stirring occasionally, until thick and syrupy, 18-20 minutes; cool.
Place 6 slices of bread on a work surface and top each with 1 1/2 ounces of the cheese. Spread about 1 rounded tablespoon of the cherry mixture over the cheese; top with 1/4 cup of the arugula and the remaining cheese and close the sandwiches with the remaining bread. Spread the outside (top and bottom) of each sandwich with the butter.
Heat a large skillet over medium-low heat. Add 2 sandwiches and cover the pan. Cook, turning once or twice, until the cheese has melted and the outside of each sandwich is golden brown and toasty, about 4 minutes per side. Repeat with remaining sandwiches. Let the sandwiches stand a minute or two before cutting them in half.