In stand mixer with paddle attachment, stir together 1 cup of the flour, the sugar, yeast and salt. Add hot water; beat on medium speed until smooth, about 2 minutes. Reduce speed to medium-low. Add eggs, 1 at a time, beating well after each addition. Gradually beat in 2 cups of the remaining flour.
Add butter, a few cubes at a time, beating until completely combined between additions. Gradually beat in 2 cups of the remaining flour to make soft sticky dough; cover with plastic wrap. Let rise in warm draft-free place until doubled in size, about 3 hours. (Make-ahead: Let rise in refrigerator for up to 24 hours.)
Gently deflate dough; cover and refrigerate for at least 6 hours or for up to 18 hours. (Make-ahead: Wrap in plastic wrap and freeze for up to 2 weeks; thaw in refrigerator for 24 hours before continuing with recipe.)
Turn out dough onto lightly floured work surface. Using no more than 1/2 cup of the remaining flour to keep hands floured during shaping, gently knead dough into ball. Divide into quarters; divide each quarter into five equally sized pieces (each about 70 g) to yield 20 pieces total. Remove 1 tsp dough from each piece; roll into 20 small balls.
Shape 1 of the large dough pieces into ball, folding ends underneath and pinching at bottom; roll on work surface in circular motion, seam side down, into smooth ball. Repeat with remaining large dough pieces. Divide large dough balls among 20 well-greased brioche moulds or wells of muffin pan. Using scissors, cut deep X in top of each; nestle 1 small ball into each X. Cover with lightly floured tea towel. Let rise in warm draft-free place until doubled in size, about 1 hour. Whisk egg yolks with 1 tsp water; brush over brioche.
Bake in 400°F oven, switching and rotating pans halfway through, until golden, 18 to 22 minutes. Let cool slightly in pans. Remove from pans; serve warm or let cool completely on rack. (Make-ahead: Let cool completely. Store in airtight container for up to 24 hours or freeze in resealable freezer bag for up to 1 week; thaw in bag at room temperature. Reheat in 325°F oven until warm, about 5 minutes.)
Tip from The Test Kitchen: For perfect, evenly sized brioche buns, be sure to weigh the dough balls with a kitchen scale.
Makes 20 brioche.