Whisk eggs in a bowl until lightly beaten. Add milk, maple syrup, and vanilla extract; gently beat until incorporated. Pour mixture into six 1/4-cup ramekins. Arrange ramekins in a baking dish. Pour enough water into baking dish to cover ramekins halfway up the sides.
Bake in the preheated oven until a knife inserted in the center of a custard comes out clean, 35 to 45 minutes. Remove ramekins from baking dish and refrigerate until completely chilled, at least 1 hour.