Heat oven to 350°F. In a small pot over medium heat, cook tomato, onion and garlic until tomato breaks down and resembles jam, 15 minutes.
Add vinegar and honey; cook until jam thickens, 15 minutes. Season with salt and black pepper; remove from heat.
On a cookie sheet, arrange bread in a single layer. Top each slice with 1/4 cup cheese and 1 slice prosciutto; bake until cheese melts, 3 minutes.
Divide tomato jam among 4 sandwich halves; top with remaining sandwich halves.
In a medium skillet over medium heat, heat 1 tablespoon oil. Cook one sandwich, flipping once, 3 minutes per side. Repeat with remaining oil and sandwiches.
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