Step 1: Prepare the Brioche Dough: In the bowl of a stand mixer fitted with the dough hook, combine the flour, sugar, salt, and yeast.
Step 2: Add Wet Ingredients: Add the lukewarm milk, beaten eggs, and vanilla extract to the dry ingredients. Mix on low speed until a shaggy dough forms.
Step 3: Incorporate the Butter: With the mixer on low, gradually add the softened butter, a few cubes at a time, allowing each addition to fully incorporate before adding more. This process may take 10-15 minutes. The dough will become very soft and sticky.
Step 4: First Proof: Transfer the dough to a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for 1-1.5 hours, or until doubled in size. Alternatively, proof in the refrigerator overnight for enhanced flavor.
Step 5: Prepare the Cinnamon-Sugar Coating: In a shallow bowl, combine the granulated sugar, cinnamon, and nutmeg (if using). Mix well.
Step 6: Shape the Dough: Punch down the risen dough and turn it out onto a lightly floured surface. Gently roll the dough into a large rectangle (about 12x18 inches).
Step 7: Cut and Coat: Using a pizza cutter or knife, cut the dough into approximately 1-inch squares. Roll each square in the cinnamon-sugar mixture, ensuring it is fully coated.
Step 8: Arrange in Pan: Grease a 10-inch bundt pan or a 9x13-inch baking pan. Arrange the coated dough pieces in the prepared pan, layering them randomly.
Step 9: Second Proof: Cover the pan with plastic wrap and let rise in a warm place for another 30-45 minutes, or until the dough has puffed up slightly.
Step 10: Prepare the Butter Sauce: While the dough is proofing, prepare the butter sauce. In a saucepan over medium heat, melt the butter. Add the brown sugar, heavy cream, and vanilla extract. Stir constantly until the sugar is dissolved and the sauce is smooth. Bring to a simmer and cook for 1-2 minutes, stirring occasionally.
Step 11: Pour Sauce Over Dough: Pour the warm butter sauce evenly over the dough in the pan, ensuring it seeps down into the layers.
Step 12: Bake: Preheat oven to 350F (175C). Bake for 30-35 minutes, or until the top is golden brown and the center is cooked through. To prevent excessive browning, tent the pan with foil during the last 10-15 minutes of baking, if needed.
Step 13: Cool and Invert: Let the baked pastry cool in the pan for 10-15 minutes before inverting it onto a serving plate. If using a 9x13 inch baking pan, serve directly from the pan.
Optional Tips: For a richer flavor, use brown butter in the butter sauce. Pecans or walnuts can be added to the cinnamon-sugar mixture for extra crunch. A cream cheese glaze can be drizzled over the finished pastry for added decadence.