Preheat the oven to 400 degrees F.
Spread the bread cubes on a large baking sheet and bake, stirring occasionally, until lightly golden on all sides, about 10 minutes. Remove from the oven.
Meanwhile in a medium skillet, heat the butter until it begins to foam, about 1 minute. Reduce the heat to low, add the rosemary and cook until fragrant, about 2 minutes. Season with salt and pepper and remove from the heat.
Alternate cubes of bread and cheese on 10-inch skewers, starting and ending with bread. Suspend the skewers over a shallow pan, making sure that the bread and cheese don't touch the bottom. Brush the rosemary butter over the bread and cheese cubes. Bake until the bread is golden and crunchy and the cheese is beginning to melt, about 10 minutes.
Remove from the oven. Slip the bread and cheese cubes off the skewers onto a warm platter and serve immediately.