Preheat the oven to 400 degrees F (200 degrees C).

Make custard: Mix eggs and brown sugar in a large bowl until smooth. Whisk in flour and salt. Slowly add milk, stirring constantly until well combined. Stir in vanilla and set aside to rest.

Prepare apples: Melt butter, brown sugar, and water in a 10 1/4-inch cast iron skillet over medium-high heat. Stir frequently until mixture starts to boil. Reduce heat to medium; swirl the skillet, but do not stir, until mixture starts to thicken, about 2 minutes. Add apples and cinnamon. Cook and stir until apples are crisp-tender and starting to brown, 15 to 18 minutes. Remove from heat.

Stir custard and pour over apples in the skillet.

Bake in the preheated oven until edges are golden brown and custard is set, 28 to 35 minutes.