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.