Combine the wine and shallots in a medium saucepan. Bring to a boil and cook for 15-20 minutes, until the mixture has reduced to about ½ cup. Whisk in the butter and sugar; remove from heat and season with salt and pepper to taste.
Heat an indoor grill to high heat. Brush the grates with oil, then place the mushrooms on the grill gill-side down. Brush the tops of the mushrooms with the cabernet sauce. Cook for about 5 minutes, then flip the mushrooms over. Brush the gill-side with additional sauce and cook for 5 minutes more. Flip the mushrooms one more time and brush the tops with the rest of the sauce. Once the mushrooms are easily pierced with a fork (the time depends on the thickness of the mushrooms, but most will be done after 15 minutes on the grill), top them each with a slice of cheese and transfer the mushrooms to a plate.
Brush the cut sides of the bread with oil and sprinkle with rosemary. Place the bread, cut-side down, onto the grill and cook until lightly toasted, about 3 minutes. Top the bottom halves of the bread with the mushrooms, tomatoes, and arugula. Cover with the top halves of the bread.
Notes
If you don't have an indoor grill, a grill pan or an outdoor grill will work instead, but cooking times will vary.
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