Adjust the oven rack to the middle position and heat the oven to 350 F. Heat 1/4 cup oil in a 12-inch ovensafe skillet over medium-low heat until shimmering. Add bread and stir to coat. Cook, stirring constantly, until bread is browned and toasted, about 5 minutes. Transfer bread to a bowl.

Return the now-empty skillet to low heat and add remaining 2 tablespoons of oil and the garlic. Cook, stirring constantly, until garlic is golden at the edges, 30 to 60 seconds. Add tomatoes, sugar, salt and pepper and stir to combine. Increase heat to medium-high and cook, stirring occasionally, until the tomatoes have started to break down and have released enough juice to be mostly submerged, 8 to 10 minutes.

Remove the skillet from the heat and gently stir in 3 cups of bread until completely moistened and evenly distributed. Using a spatula, press down on the bread until it is completely submerged. Arrange remaining 1 cup of bread evenly over surface, pressing to partially submerge. Sprinkle evenly with Parmesan.

Bake until the top of the gratin is deeply browned, tomatoes are bubbling, and juice has reduced, 40 to 45 minutes; after 30 minutes, run a spatula around the edge of the skillet to loosen the crust and release any juice underneath. (Gratin will appear loose and jiggle around outer edges but will thicken as it cools.)

Remove the skillet from the oven and let stand for 15 minutes. Sprinkle gratin with basil and serve.