With a food processor running, drop in garlic until minced. Turn off machine and add tomatoes. Pulse until finely chopped. (This step can also be done manually if preferred, which is how I did it).
Place mixture in a fine sieve over a bowl, stir in salt and allow to drain 30 minutes, pressing down on tomatoes from time to time.
Transfer tomatoes to a bowl and stir in basil and breadcrumbs. Season to taste with salt and pepper.
Arrange half the bread slices on a work surface and spread with tomato mixture. Cover with a thin layer of the cheese and top with remaining bread.
Film a large non-stick skillet with oil. Place as many sandwiches as will fit comfortably and fry over medium-low or medium heat, turning to brown both sides and pressing down lightly with a spatula until golden on both sides. (About 1 1/2 - 2 minutes per side.) Remove to paper towel to drain.
Repeat if necessary while finished sandwiches are kept in a 200 degree Fahrenheit oven to be kept warm. Quarter sandwiches on the diagonal for canapes.