Prepare the Dough: Bring the eggs and unsalted butter to room temperature. Combine 1 cup of flour, active dry yeast, granulated sugar, and salt in a stand mixer bowl. Gradually add warm water, mixing until smooth. Incorporate eggs one at a time.

Add Flour and Butter: Gradually add 2 cups of all-purpose flour, mixing to form a cohesive dough. Add softened butter in 2–3 batches, allowing full absorption each time.

Incorporate Remaining Flour: Add the remaining flour until the dough is thick and slightly sticky. Switch to a dough hook if needed.

First Rise: Cover and let the dough rise at room temperature for about 3 hours until doubled. Punch down, cover, and refrigerate overnight.

Preheat and Shape: Preheat oven to 350°F. Divide the chilled dough into two equal pieces, shape into logs, and place in greased loaf pans.

Second Rise: Cover and allow to proof for 2½ to 3 hours until doubled.

Bake the Brioche: Bake for 37 to 40 minutes until golden brown. Brush with sugar water during the last few minutes for extra shine.

Cool and Serve: Let cool in pans for 5 minutes, then transfer to a wire rack. Cool completely before slicing.