Combine ¼ cup warm water and warm milk in bowl of heavy-duty mixer fitted with paddle attachment. Sprinkle yeast over and stir to moisten evenly. Let stand until yeast dissolves, stirring occasionally, about 8 minutes.
Add flour and salt to yeast mixture. Blend at medium-low speed until shaggy lumps form, scraping down sides of bowl occasionally, 1 to 2 minutes. Add eggs, 1 at a time, beating until blended after each addition. Beat in sugar. Increase mixer speed to medium; beat until dough is smooth, about 3 minutes.
Reduce speed to low. Add butter, 1 tablespoon at a time, beating until blended after each addition, about 4 minutes (dough will be soft and silky). Increase speed to medium-high and beat until dough pulls away from sides of bowl and climbs paddle, 8 to 9 minutes.
Lightly butter large bowl. Scrape dough into bowl. Cover bowl with plastic wrap. Let dough rise in warm draft-free area until almost doubled in volume, about 1 hour 15 minutes to 1 hour 30 minutes.
Gently deflate dough by lifting around edges, then letting dough fall back into bowl, turning bowl and repeating as needed. Cover bowl tightly with plastic wrap and chill, deflating dough in same way every 30 minutes until dough stops rising, about 2 hours. Chill overnight. (At this point, use the dough to make 12 brioches, or 6 brioches and 1 tart, or 2 tarts.)
Butter 12 standard (1/3-cup) muffin cups. Divide dough into 12 equal pieces; cut each piece into thirds. Roll each small piece between palms into ball. Place 3 balls in each prepared cup (dough will fill cup).
Place muffin pan in warm draft-free area; lay sheet of waxed paper over. Let dough rise until light and almost doubled (dough will rise ½" to 1" above top rim of muffin cups), 50 to 60 minutes.
Position rack in center of oven and preheat to 400°F. Place muffin pan on rimmed baking sheet. Gently brush egg glaze over risen dough, being careful that glaze does not drip between dough and pan (which can prevent full expansion in oven).
Bake brioches until golden brown, covering with foil if browning too quickly, about 20 minutes. Transfer pan to rack. Cool 10 minutes. Remove brioches from pan. Serve warm or at room temperature.