Loosen the yogurt first. Whisk the Greek yogurt and milk together in a wide shallow bowl until completely smooth. Do this before the eggs go anywhere near it.
Add the eggs, cinnamon and vanilla. Whisk until the custard is one uniform colour with no streaks of white.
Soak the bread 25 to 30 seconds per side. Day-old sourdough takes about that long. Press it gently into the custard rather than letting it float.
Heat a skillet over medium-low and melt half the butter. Medium-low, not medium — this matters, and the reason is in the tips.
Cook 3 to 4 minutes per side, until deeply golden. Add the remaining butter when you flip.
Rest 1 minute before serving with the maple syrup.