Preheat the oven to 425°F (220°C) and let it come fully up to temperature before the glass dish goes in.
As soon as the oven starts preheating, warm the Greek yogurt, milk, and maple syrup together in a microwave safe bowl for 30 seconds, then stir until smooth.
Add the warm yogurt mixture to a food processor or blender along with the eggs, flour, vanilla, and salt, and blend for about 30 seconds until completely smooth with no streaks of flour, scraping down the sides once if needed. Let the batter sit on the counter while the oven finishes preheating and the butter browns.
Cut the butter into pieces, place them in the 9x13 glass baking dish, and set the dish in the hot oven until the butter melts and the milk solids turn golden brown, about 8 to 10 minutes, watching closely near the end so it browns without burning.
Carefully pull out the oven rack and slowly pour the batter into the center of the dish, letting it spread toward the edges on its own rather than pouring it against the hot sides.
Bake for 20 to 25 minutes without opening the oven door, until the edges are deeply golden and puffed up the sides of the dish and the center is set.
Set the hot dish on a dry towel or trivet, never a wet or cold surface, then drizzle with maple syrup, slice into 6 pieces, and serve right away while it is still puffed.