Japanese Milk Bread

FOR THE ROUX
1/2 cup water
1/4 cup plus 2 tablespoons whole milk
1/4 cup (34 grams) bread flour

FOR THE DOUGH
113 grams (8 tablespoons) salted butter, cut into 8 pieces, room temperature, plus 28 grams (2 tablespoons), melted, for brushing
3 large eggs, divided
1 cup whole milk, room temperature
639 grams (4 2/3 cups) bread flour, plus more for dusting
60 grams (1/2 cup) rye flour
80 grams (1/4 cup plus 2 tablespoons) white sugar
27 grams (1/4 cup) nonfat or low-fat dry milk powder
1½ tablespoons instant yeast
1¾ teaspoons table salt