Adjust 1 oven rack to lowest position and second rack 5 to 6 inches from broiler element. Heat oven to 425 degrees. Spray baking sheet well with vegetable oil spray.
In large bowl, whisk eggs, vanilla, brown sugar, cinnamon, and salt until well combined and sugar is dissolved, about 30 seconds. Add milk and melted butter and whisk until combined.
Pour egg mixture into greased baking sheet.
Place bread slices in 2 rows on baking sheet. Working quickly, flip each slice in same order you placed them on baking sheet. Let bread sit until slices absorb custard, about 1 minute.
Place baking sheet in oven. Bake on lower rack until bottoms of slices are golden brown, 10 to 15 minutes.
Ask an adult to use oven mitts to transfer baking sheet to upper rack. Heat broiler and broil until tops of slices are golden brown, 1 to 4 minutes (watch carefully to prevent burning!).
Ask an adult to remove baking sheet from oven and place on cooling rack. Serve.
