Preheat the oven to 350 degrees F (175 degrees C).

Cook and stir 1/2 cup sugar and 3 tablespoons butter in a saucepan over low heat until butter is melted, 3 to 4 minutes. Stir in cream; bring to a boil, reduce heat, and simmer until sauce is slightly thickened, 5 to 10 minutes. Transfer sauce to a bowl; cover with plastic wrap. Set aside

Beat 3/4 cup sugar and 3 tablespoons softened butter together in a bowl using an electric mixer until smooth and creamy. Add eggs and vanilla; beat until fluffy, 2 to 3 minutes. Slowly stir in milk, while constantly stirring, until evenly combined.

Place bread cubes in a large bowl; pour milk mixture over bread. Let stand until bread is moistened, about 5 minutes. Fold in blueberries. Transfer to a 3-quart baking dish.

Combine 2 tablespoons sugar and cinnamon in a bowl; sprinkle over bread mixture. Drizzle 1/3 cup sauce over bread mixture.

Bake in the preheated oven until a knife inserted in the center comes out clean, about 35 minutes. Serve with remaining sauce.