Position an oven rack 6 inches from the broiler element and heat the broiler on high; line a sheet pan with foil.
In a medium bowl, combine the tomatoes, kosher salt, red wine vinegar, black pepper, and 3 tablespoons of the extra-virgin olive oil; fold in the fresh basil leaves. Let stand 15–20 minutes so the juices bead and flavors meld.
Arrange the country bread on the prepared pan and brush both sides with the remaining 3 tablespoons oil.
Broil the bread until golden with light char at the edges, 2–3 minutes per side, watching closely to prevent burning.
While hot, rub one side of each slice with the cut side of the garlic clove.
Stir the tomato mixture and spoon it generously over the rubbed side of each toast, letting a little juice soak in.
Transfer to a platter and serve immediately while the bread is crisp and the topping is fresh.