Mix the dough: Combine the flour, yeast and salt in the bowl of a stand mixer, then add the eggs, milk, maple syrup and vanilla and mix with the dough hook on low speed until a rough dough forms.
Begin kneading: Increase the speed to medium-low and knead for about 5 minutes, until the dough begins to develop some strength and elasticity.
Add the butter: With the mixer running on medium-low, add the softened butter a few pieces at a time, allowing each addition to incorporate before adding more.
Finish kneading: Continue kneading for another 8–12 minutes, until the dough is smooth, glossy and elastic and a small piece can be stretched into a thin membrane without immediately tearing.
Let the dough rise: Transfer the dough to a lightly buttered bowl, cover and let rise somewhere comfortably warm for 1½–2½ hours, or until puffy and roughly doubled.
Chill overnight: Gently press down the dough, cover the bowl tightly and refrigerate for 8–16 hours.
Prepare the pan: Butter a 9 × 5-inch loaf pan and line it with parchment paper, leaving some overhang along the long sides.
Roll the dough: Place the cold dough on a lightly floured surface and roll it into an approximately 8 × 14-inch rectangle.
Add the filling: Spread the softened butter over the dough, leaving a ½-inch border around the edges, then sprinkle the maple sugar evenly over the top.
Shape the loaf: Starting at one of the 8-inch ends, roll the dough into a snug log, pinch the seam closed and place it seam-side down in the prepared pan.
Let it rise again: Cover loosely and let rise for 2–4 hours, or until very puffy and the centre of the loaf sits about 1 inch above the rim of the pan.
Check the dough: Gently press the dough with one finger; it should slowly spring back while leaving a slight indentation.
Preheat the oven: Preheat the oven to 350°F (175°C).
Add the egg wash: Whisk together the egg and milk, then gently brush the mixture over the risen loaf.
Bake: Bake for 35–45 minutes, tenting the loaf loosely with foil if the top browns too quickly, until the centre reaches 190–195°F (88–91°C).
Remove from the pan: Let the brioche rest in its pan for 10 minutes, then lift it out and transfer it to a cooling rack.
Add the maple finish: Melt the butter and stir in the maple syrup and salt, then brush the mixture lightly over the warm loaf.
Cool: Let the brioche cool for at least 1 hour before slicing.