Heat oven to 356°F (180 °C); prepare medium tray.
Wash the mushrooms or rather, scrub them with your palms. Remove the stems. Dry each mushroom with paper towel on both sides.
Melt the butter in a casserole with the pressed garlic. Using a brush, scrub each mushroom on its top and bottom side. Arrange in the tray. Place a piece of chopped tomato, piece of mozzarella in place of the stem. Salt them. Bake until the mozzarella melts, about 15 min. Take out of the oven and pour a trickle of glaze on top.
Pour the balsamic vinegar with the sugar in a casserole. Leave on moderate heat while stirring, until your spoon leaves a furrow along the bottom, about 8-10 min. The sauce needs to be thick with a pleasant dark brown-caramel color and have a sweet-and-sour taste with strong aroma. Pour it over each mushroom in a thin trickle.
Notes: Portobello mushrooms are large, flat, fleshy and brown. They have nearly minimal differences taste-wise in comparison to plain champignons.
I've used champignons before in place of Portobellos for this recipe and it's fine.