Using a small serrated knife, cut off the top eighth of each tomato and hollow out the insides, leaving the sides intact. (Discard the seeds and insides.) Slice a small sliver off each tomato bottom so it sits flat. Drain well on paper towel, taking care to dry the inside.
Sprinkle the tomatoes with 1/4 teaspoon salt. Stuff each tomato with a pinch of the chopped basil and a mozzarella ball and set on a small rimmed baking sheet.
Combine the breadcrumbs with the olive oil, oregano and the remaining 1/2 teaspoon salt in a small bowl. Divide the mixture among the tops of the tomatoes.
Bake until golden and the cheese is melted, about 10 minutes. Allow to cool for 10 minutes, then serve warm.