1. Adjust two oven racks to the top third and bottom third of the oven. Preheat oven to 400 degrees F. Toss the bagel cubes with the melted butter and spread on 2 large baking sheets. Bake 10 to 15 minutes or until dried and toasted, rotating pans once halfway through baking.
2. Coat a 3-qt. baking dish with butter. Cover the bottom of the dish with one-third of the bagel cubes. Sprinkle on one-third each of the cheese, bacon, and green onions. Repeat layers twice, starting with the bagel cubes and ending with the green onions.
3. In a large bowl, whisk together the eggs, milk, garlic, salt, and pepper. Pour the mixture over the casserole, making sure to moisten all the bread. Use a sheet of plastic wrap to gently press the bread into the egg mixture and cover the casserole. Chill at least 2 hours or up to overnight.
4. When ready to bake, preheat the oven to 350 degrees F. Remove the strata from the refrigerator and let stand at room temperature for 30 minutes. Bake until the strata is set in the center (160 degrees F), about 50 minutes, covering with foil during the last 20 minutes, if needed to prevent overbrowning. If desired, sprinkle with additional green onions just before serving.