Mix together the unsifted flour, 3 tablespoons sugar, 1-1/2 teaspoons salt, yeast, water, and butter in the bread machine on the manual setting.
OR, mix 1-1/2 cup flour, 3 tablespoons sugar, 1-1/2 teaspoons salt, and yeast. Heat water and butter to 120-130 degrees F. Add water to dry ingredients; beat 2 minutes at medium speed on mixer. Add 1/2 cup flour, beat at high speed 2 minutes. Stir in more flour to make a stiff dough. Knead 8-10 minutes. Cover and let rise. Punch down. Cover; let rest 15 minutes.
Remove from bread machine.
Divide dough into 12 pieces; shape 3 pieces into smooth balls. Poke a 1 inch hole in each. Drop bagels into a large skillet with 1 inch water (simmering) with 1 tablespoon sugar and 2 teaspoons salt in the water. Cook on medium low heat for 3 minutes, turn and cook 2 minutes; turn again, cook 1 minute more. Drain on towels. Repeat for rest of dough. Place on greased cookie sheet.
Bake at 375 degrees F (190 degrees C) for 20-25 minutes. Remove from sheets and cool. If desired, before baking, mix 1 egg white and 1 tablespoon water, brush on bagels, and sprinkle with sesame, poppy, or caraway seeds.