Combine warm milk and yeast in the mixer bowl. Let sit 10 minutes.
Mix in flour and salt with paddle attachment of mixer (you will have a shaggy dough), then work in the eggs, one at a time, along with 2 tablespoon maple sugar.
Knead the mixture to form a smooth, elastic dough for about 10 minutes with the dough hook.
Switch back to the paddle, and beat in the cubes of butter. When the dough starts to take shape again, you can switch back to the hook and knead for about 8 minutes. You will regain that smooth, elastic dough that stretches thin without tearing.
Transfer the dough to a buttered bowl and let sit for 3 hours, covered with saran wrap. The dough will double in size.
Punch down the dough, cover the bowl again and refrigerate overnight.
Roll the dough out on a lightly floured surface to a 12x12" square. Brush with 1 tablespoon melted butter and sprinkle with ~2 tablespoon maple sugar.
Cut out sixteen circles (about 3–3.15 inch diameter)
Place 3 of the circles side-by-side, overlapping them as in the step-by-step photo above. Roll the circles together into a log and cut in half. Place each half into greased muffin tins, cut-side down. Repeat with remaining circles. This way you will make 10 buns. For the last two brioche buns, you will have to cut out half circles from the leftover rolled out dough fitting the cookie cutter where you can and repeat the rolling process. I saved the dough scraps and placed them in a greased mini loaf pan because one should never waste brioche dough.
Cover the buns with clean tea towels and let proof at room temperature for about an hour to puff a little. Meanwhile, preheat the oven to 350 °F (175 °C).
Combine the 2 tablespoon melted butter with 2 tablespoon maple syrup and brush this over all the buns, generously. Bake in the preheated oven for about 23 minutes. Unmold quickly so they don't get stuck in the pan!
Eat them all straightaway.
Notes
Make sure you use pure maple sugar (like this one on Amazon)