Combine 1/2 cup water, sugar, and yeast in a large bowl. Let stand 5 minutes or until mixture is bubbly. Weigh or lightly spoon white whole-wheat flour into dry measuring cups; level with a knife. Stir 1 cup white whole-wheat flour, milk, butter, salt, and egg into yeast mixture. Weigh or lightly spoon all-purpose flour into dry measuring cups; level with a knife. Gradually add remaining 1/2 cup white whole-wheat flour and 2 cups all-purpose flour to yeast mixture, stirring until a soft dough forms.
Sprinkle 2 tablespoons all-purpose flour on a work surface. Turn dough out onto surface; knead until flour is incorporated. Knead in remaining 2 tablespoons all-purpose flour until dough is smooth and elastic (about 10 minutes). Place dough in a large bowl coated with cooking spray, turning to coat. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size.
Preheat oven to 400°.
Coat 18 muffin cups with cooking spray. Punch dough down; turn out onto a lightly floured surface. Cover and let rest 15 minutes. Divide dough into 18 pieces. Roll each piece into a 1-inch-thick rope; divide each into 3 pieces. Roll each piece into a ball. Place 3 balls into each of the prepared muffin cups; cover and let rise 30 minutes or until doubled in size. Bake at 400° for 18 minutes or until golden brown.