Preheat the oven to 350°F (175°C).
In a mixing bowl, whisk together the milk, sugar, vanilla extract, and eggs.
Arrange the bread slices in a baking dish and pour the milk mixture over them. Let it soak for 5-10 minutes.
Bake in the preheated oven for 25-30 minutes, or until the custard is set and the top is golden.
Meanwhile, in a small saucepan, combine the blackberries and maple syrup over medium heat.
Cook until the blackberries are soft and the syrup is slightly thickened, about 5 minutes.
Serve the baked French toast with the blackberry maple syrup drizzled on top.