Place the bread slices on a baking sheet fitted with a rack. Brush both sides of each slice with a bit of olive oil. Toast in the oven until the bread is crispy but still slightly tender on the inside, about 8 minutes. Let cool to room temperature.
Stand a box grater up in a large bowl. Using the large holes on the grater, grate the tomatoes into the bowl. Take the end pieces and chop them up. Place a mesh strainer over a bowl and strain the grated tomato pulp from the juice. Place the chopped end pieces and strained pulp into another bowl. (Reserve the tomato water for another use; it can be added to any tomato sauce, and is good in a Bloody Mary.) To the pulp, add 1/4 cup of the olive oil and the basil, and season with salt and pepper.
Top each bread slice with a generous amount of the tomato mixture. Drizzle each with the remaining olive oil. Transfer to a serving tray and serve.