1/4 cup minced shallots
9 tablespoons unsalted butter, room temperature
2 teaspoons golden brown sugar
1 tablespoon minced fresh rosemary
1 1/2 pounds ground beef
1/2 teaspoon ground black pepper
1 cup (packed) coarsely grated extra-sharp white cheddar cheese
4 4 1/2-inch squares focaccia, cut horizontally in half
8 large tomato slices
2 cups arugula
Boil wine and shallots in medium saucepan until reduced to 3/4 cup, about 20 minutes. Add 1 tablespoon butter and brown sugar; whisk until butter melts and sugar dissolves. Remove from heat. Mix remaining 8 tablespoons butter and rosemary in small bowl. Set aside.
Prepare barbecue (medium-high heat). Mix beef, salt, pepper, and 1/4 cup wine-shallot mixture in bowl. Form meat into four 5-inch squares or rounds. Brush grill rack with oil. Grill burgers until brown on bottom, about 3 minutes. Turn burgers and brush with wine-shallot mixture. Continue grilling burgers until cooked to desired doneness, turning and brushing occasionally with wine-shallot mixture, about 4 minutes longer for medium-rare. Sprinkle with cheese after last turn and grill until cheese melts.
Spread cut sides of bread with rosemary butter. Grill, cut side down, until golden, about 2 minutes. Arrange bread, grilled side up, on plates. Top bottom halves with burgers, then tomatoes and arugula. Cover with top halves of bread.