Drain the capers if packed in vinegar; soak, rinse, and drain if packed in salt; then chop them fine.
Turn the oven on to 400°.
Remove the salmon’s skin, remove any loose membranes, and carefully pick out all bones. Cut the fish into very small dice and put it in a bowl together with 2½tablespoons of olive oil, the chopped parsley, garlic, capers, 1tablespoon of bread crumbs, salt, and several grindings of black pepper. Toss thoroughly.
Wash the tomatoes, cut them in half horizontally, scoop out all the seeds and the centers to make a cup-like hollow. (If you are cooking anything else that day or the next that calls for fresh tomatoes, use the scoopings in the recipe.)
Pat the inside of the tomatoes with paper towels to soak up excess juice, then stuff them with the salmon mixture, pressing it down lightly as you do so. There should be enough to form a mound. Sprinkle the tops with the remaining bread crumbs and drizzle with the remaining olive oil, holding back a few drops to smear the bottom of a baking pan.
Place the tomatoes on the baking pan and bake in the upper level of the preheated oven for 35 minutes or until the salmon stuffing has formed a light golden crust. Serve not piping hot, but lukewarm. They are also good later, at room temperature, but not reheated.