Process the bread cubes in a food processor until it is transformed into coarse crumbs. Heat a tablespoon and a half of oil in a skillet over medium high heat. Toast the crumbs until golden brown, stirring frequently. Let drain on paper towels and sprinkle with a pinch of salt.
Turn on a grill to high heat or your broiler. Lightly coat or spray corn and put on grill or under the broiler until lightly charred, about 5-7 minutes. Turn several times to ensure even color. Set aside until cool enough to handle.
Meanwhile, chop the tomatoes into 1-inch cubes and layer on a large serving plate. Cut corn kernels off the cob and layer on top of the tomatoes. Sprinkle vegetables with salt.
Set burrata on top of the corn and tomato, and drizzle rest of the oil on top of the cheese. Sprinkle cheese and vegetables evenly with toasted breadcrumbs.
Stack basil leaves and roll tightly into a long cylinder. Slice into thin strips (chiffonade) and garnish the top of the burrata with the basil.
Sprinkle freshly ground pepper over the salad.