Whisk together eggs, milk, heavy cream, 3 tablespoons sugar, and rum. Set aside.
Spread top of each slice of bread with generous amount of butter. Cut slices in half diagonally. In a 9-inch-round baking dish or cake pan, fan overlapping slices in one layer. It is also fine for bread to overlap randomly. Pour egg mixture over bread, and let stand 1 hour.
Heat oven to 400 degrees. Sprinkle pudding with 2 tablespoons sugar. Bake until puffed and golden brown, and pudding has set, about 20 minutes.
Meanwhile, combine strawberries with remaining 2 tablespoons sugar. Let stand while pudding bakes. Serve strawberries with baked pudding.