Combine warm water and warm milk in mixing bowl fitted with paddle attachment. Sprinkle yeast over liquid and gently stir. Let stand until yeast dissolves. Add flour and salt to yeast mixture. Blend at medium-low speed for 2 minutes scraping down sides of bowl. Add eggs, one at a time, beating until blended after each addition. Mix in sugar. Increase mixer speed to medium and beat until dough is smooth, about 3 minutes. Reduce speed to low. Add butter, one tablespoon at a time, beating until blended after each addition, until dough is soft and silky, about 4 minutes. Increase speed to medium-high and beat until dough pulls away from sides of bowl and climbs paddle. Lightly butter large bowl. Place dough into bowl and cover with plastic wrap. Let dough rise in warm area until doubled in volume, about 1 hour 30 minutes. Lift dough around edges to deflate, then let dough fall back into bowl. Turn bowl and repeat. Cover bowl tightly with plastic wrap and chill. Repeat deflating process one or two more times during the next two hours while dough is still rising. Chill overnight. When ready to bake, grease 12 fluted individual brioche molds (approximately 3-inch) or large muffin cups. Pat the chilled dough into a 12×6-inch rectangle. Cut dough into 12 equal portions. From each remove a small piece and roll into 12 (1/2-inch balls.) Roll remaining portions into 12 larger balls; place in prepared pans. Using your fingers, make a deep indentation in the center of dough ball. Brush with water. Press small balls into indentations. Cover; let rise in a warm place until double in size, about 45 minutes. Position rack in center of oven and preheat to 400 degrees. Place muffin pan on rimmed baking sheet. Gently brush egg wash over risen dough, being careful that glaze does not drip between dough and pan. Bake brioches until golden brown, about 20 minutes. (Cover with foil if browning too quickly.) Transfer pan to cooling rack. Cool for about 10 minutes before removing brioches from pan. Serve warm or at room temperature.