Add flour, sugar, yeast, and salt to a large bowl. Mix well. Add warm milk, eggs, and vanilla. Stir until a soft dough forms. Add softened butter and knead until smooth.

Transfer the dough onto a lightly floured surface. Knead for about six minutes. The dough should feel soft and elastic. Avoid adding too much flour, because brioche should stay tender.

Place the dough inside a lightly greased bowl. Cover it with a clean kitchen towel. Let it rise for about one hour. It should become noticeably puffy and almost double in size.

Peel the apples if desired. Remove their cores and slice them thinly. Place the slices inside a bowl. Add brown sugar, cinnamon, nutmeg, lemon juice, and flour.

Gently toss the apples with the seasonings. Make sure every slice gets lightly coated. Let the mixture rest for five minutes. This allows the flavors to blend together nicely.

Add sugar to a small saucepan. Heat it over medium heat. Let the sugar melt slowly. Stir gently once it begins melting. Continue until it becomes a deep golden caramel.

Carefully add the butter to the melted sugar. Stir until smooth. Slowly pour in the heavy cream. The mixture may bubble strongly. Keep stirring until the caramel becomes smooth and creamy.

Remove the caramel from the heat. Stir in vanilla and a small pinch of salt. Let the sauce cool slightly. It will become thicker as it cools.

Place the risen dough on parchment paper. Roll it into a rough circle. Aim for about twelve inches wide. Do not worry about perfect edges. Rustic shapes make galettes look beautiful.

Leave a two-inch border around the dough. Arrange the apple slices in overlapping circles. Keep the slices fairly even. Avoid adding too much liquid from the apple mixture.

Drizzle about three tablespoons of caramel over the apples. Spread it gently with a pastry brush. Keep most of the caramel away from the dough edges. This helps prevent sogginess.

Carefully fold the dough edges toward the apples. Work around the circle slowly. The center should remain uncovered. Gently press any cracks together with your fingers.

Whisk one egg with one tablespoon of water. Brush the exposed dough edges lightly. Sprinkle coarse sugar around the border. This creates a beautiful golden crust.

Place the prepared galette in the refrigerator. Chill it for about fifteen minutes. This helps the butter firm up again. It also helps the brioche keep its shape.

Preheat your oven to 375°F. Place the galette on a baking sheet. Bake for about 30 to 35 minutes. The crust should look deeply golden and the apples should be tender.

Remove the galette from the oven. Let it cool for at least fifteen minutes. This allows the caramel to thicken slightly. It also makes slicing much easier.

Warm the remaining caramel gently if needed. Drizzle it over the baked apples. Use a light hand. Too much caramel can hide the beautiful apple layers.

Slice the galette into eight portions. Serve it warm or slightly cooled. Add powdered sugar if desired. A scoop of vanilla ice cream makes this dessert even better.