Preheat the oven to 250 degrees F.
Arrange the tomatoes, cut-side up, in a single layer on a parchment or foil-lined baking sheet. Sprinkle with salt. Bake until slightly shriveled but still plump, about 5 hours. Layer them with the herb sprigs in a container, add olive oil to cover, seal tightly and refrigerate for up to 2 weeks.
Preheat the oven to 400 degrees F.
Brush both sides of the bread slices with some of the flavored olive oil and place on a baking sheet. Bake until golden, about 7 minutes, turning once about halfway through. While the toasts are hot, rub both sides with the whole garlic clove
, placing them back on the baking sheet as you work. Leave the oven on.
Put a tomato and a small slice of cheese on each toast - it's yummy either with the tomato on the cheese or with the cheese on top. Bake just until cheese starts to melt
, about 3 minutes. Serve warm.