1Place apples in a bowl and drizzle with lemon juice to keep from browning.
2Melt 1/2 cup butter in a large saucepan over medium heat; remove from heat. Stir flour into melted butter until smooth. Return saucepan to low heat; stir in brown sugar, white sugar, and water until sugars dissolve and mixture is smooth, 3 to 5 minutes. Increase heat to medium; cook and stir mixture until boiling. Reduce heat to low and add apples. Simmer mixture until apples are cooked but slightly tender, 20 to 25 minutes.
3Preheat griddle to 325 degrees F (165 degrees C).
4Whisk eggs, milk, cinnamon, and vanilla extract together in a bowl. Dip bread slices into egg mixture until completely coated.
5Grease heated griddle with frozen butter. Cook dipped bread slices on the hot buttered griddle until browned, 2 to 3 minutes per side. Repeat with remaining frozen butter and dipped bread. Serve French toast with apple mixture ladled over top.
For varied textures, use more than one kind of apple. Depending on what variety of apple you use, time for cooking will differ.