Coat a 9 x 13 baking dish or casserole dish with cooking spray. Cut the bagels into bite-sized pieces and place in baking dish.
In a large bowl, whisk together the eggs, milk, melted butter, brown sugar, cinnamon, vanilla, and salt. Pour the mixture over the bagels and press down gently to soak all of the pieces. Cover and refrigerate overnight or for at least 4 hours.
Preheat oven to 350 degrees. Bake, uncovered, for 30-35 minutes until the custard has set in the middle.
While baking, make the glaze topping by whisking all glaze ingredients together in a small bowl, adding up to an additional tablespoon of milk if needed to made glaze pourable. Drizzle over the baked bagels. Serve with warm maple syrup.
