Preheat the oven to 400 degrees . In a medium skillet, melt 1 1/2 tablespoons butter over medium heat. Add the onion, season with salt and pepper and cook until softened, about 5 minutes. Add the ham and cook for 2 minutes more.
Quarter 4 brioche slices and arrange in a buttered 8-inch square baking dish, overlapping slightly. Layer the turkey, cheese and onion-ham mixture on top.
Cut the remaining 3 brioche slices into 3/4-inch cubes. In a glass bowl, melt the remaining 1 1/2 tablespoons butter in the microwave; add the cubed brioche and toss. Arrange evenly on top of the casserole.
In the same bowl, whisk together the eggs, milk and 1/4 teaspoon salt. Pour over the casserole, then let stand for 20 minutes.
Bake until set in the center, about 30 minutes; cover with foil for the last 10 minutes to prevent the top from browning too much. Let stand for 10 minutes. Serve with the syrup.