Step 1: In the bowl of a stand mixer, combine warm milk, 1 teaspoon granulated sugar, and yeast. Let stand for 5-10 minutes until foamy.
Step 2: Add flour, remaining 1/4 cup granulated sugar, and salt to the yeast mixture. Add eggs, one at a time, mixing on low speed until combined.
Step 3: Increase speed to medium and knead for 8-10 minutes until the dough is smooth and elastic. The dough will be sticky.
Step 4: With the mixer still running on medium-low, add cold butter cubes, one or two at a time, allowing each addition to be fully incorporated before adding more. This will take about 10-15 minutes. The dough will become very soft and shiny.
Step 5: Transfer the dough to a lightly oiled bowl, cover with plastic wrap, and refrigerate for at least 6 hours, or preferably overnight, for its first rise.
Step 6: While the dough chills, prepare the apple filling. In a medium saucepan, melt 2 tablespoons butter over medium heat. Add diced apples, brown sugar, cinnamon, nutmeg, and lemon juice. Cook, stirring occasionally, for 8-10 minutes, until apples are tender but still hold their shape.
Step 7: Stir in the cornstarch slurry and cook for another 1-2 minutes until the filling thickens. Remove from heat and let cool completely.
Step 8: On a lightly floured surface, gently roll out the chilled brioche dough to about 1/2-inch thickness. Using a 3 to 3.5-inch round cutter, cut out donuts. Gently gather and re-roll scraps once or twice to cut more donuts.
Step 9: Place the cut donuts on parchment-lined baking sheets, leaving space between them. Cover loosely with plastic wrap and let proof in a warm, draft-free place for 1-1.5 hours, or until visibly puffy and light.
Step 10: In a large heavy-bottomed pot or Dutch oven, heat vegetable oil to 350-360°F (175-180°C).
Step 11: Carefully lower 2-3 donuts into the hot oil. Fry for 2-3 minutes per side, until golden brown. Adjust heat as needed to maintain oil temperature.
Step 12: Remove donuts with a slotted spoon and place on a wire rack lined with paper towels to drain excess oil. Allow to cool slightly.
Step 13: In a shallow dish, combine 1/2 cup granulated sugar and 1 tablespoon cinnamon. While donuts are still warm, roll them in the cinnamon-sugar mixture to coat.
Step 14: Once donuts are cooled, use a piping bag fitted with a long, thin pastry tip to inject the cooled apple pie filling into the side of each donut until plump. Serve immediately and enjoy these delicious Apple Pie Brioche Donuts: A Must-Try Cozy Treat!
Step 15: Store any leftover donuts in an airtight container at room temperature for up to 2 days.