Heat the oven to 350 degrees F.
Adjust the rack to the center. Generously butter a 13 by 9 by 2-inch baking dish. Set aside. In a small bowl, combine the flour, baking powder, and salt. Set aside. In a large bowl, whisk together the eggs and the milk. Whisk the flour mixture into the eggs, until the batter is smooth. Whisk in the cheese; the batter will be clumpy and look curdled.
Using tongs, dip the brioche slices into the batter, coating well. Arrange the slices side by side in the buttered pan. Set aside.
Stir the diced tomatoes and chopped basil into the remaining batter. Pour the batter over the brioche slices.
Bake the French toast until golden brown, 30 to 40 minutes. Let cool for 5 to 10 minutes before cutting. Invert onto a cooling rack and turn right side up onto a cutting board. Cut into desired shapes or sizes using a knife or cookie cutter. Garnish with remaining basil strips, and serve warm.
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