Combine water, sugar, and yeast in a large bowl or the bowl of a standing electric mixer. Let sit until bubbly, 5 to 10 minutes.
Add milk, honey, salt, oil, and 2 cups of flour to the yeast mixture. Mix well with a wooden spoon or the mixer's paddle attachment. Add remaining flour about a cup at a time until a shaggy dough pulls away from the sides of the bowl.
Turn the dough out on a floured surface or switch to mixer's dough hook. Knead in remaining flour until the dough is smooth and slightly tacky, about 5 minutes.
Transfer the dough to a deep oiled bowl, turning it once to coat the top with oil. Cover the bowl with plastic wrap and let the dough rise in a warm place until doubled in bulk, about an hour.
Turn the dough out on a lightly floured work surface, pressing on it gently with your fingertips to deflate it. Divide it into 2 equal portions, shape into balls, cover, and let rest for 10 minutes. In the meantime, grease two 9 by 5-inch loaf pans.
Working one at a time, flatten the dough into rectangles slightly smaller than the pans. Roll each rectangle up tightly, pinching the seams together to seal, and place the dough, seam side down, in loaf pans. Cover with plastic wrap and let rise until doubled in bulk, about 40 minutes.
Preheat oven to 375º F. In a small bowl, lightly beat egg and milk together for the glaze.
Brush the loaves with the glaze and bake in the middle of the oven until golden brown and pulling away from the sides, about 40 minutes.
Turn bread out of pans and cool on a rack for at least 30 minutes before slicing.