Following manufacturers directions, add butter, eggs, milk and honey to bread machine. Top with flours, sugar and salt. Make well in center of dry ingredients with back of spoon, without breaking through to liquid. Add yeast to well, being careful it does not come into contact with the liquids. Close lid of machine.
Set bread machine on dough setting. Begin cycle; scrape down sides if necessary through cycle. When cycle is finished, remove dough to lightly floured surface. Punch dough down lightly; let rest 2 minutes.
Meanwhile, coat two 5-cup brioche pans (Note, below) with cooking spray. Divide dough in half. Snip 2 tablespoons dough from each half. Shape each remaining portion of dough into smooth ball; place one in each prepared pan. Make dimple in center top of each of doughs. Place small ball in center of each indentation. Cover loosely with clean kitchen towel; let rise in warm place 1 1/4 hours, until almost doubled.
Heat oven to 375 degrees F. Brush lightly beaten egg over top of loaves. Bake in 375 degree F oven 20 minutes, until golden and puffed. Remove loaves from pans to racks. Let cool 15 minutes. Serve slightly warm or let cool completely. Makes 2 loaves (6 slices each).