Using the bread machine: Mix together 560g flour, 3 tablespoons sugar, 1 1/2 teaspoons salt, yeast, water and butter in the bread machine on the manual setting. Remove from bread machine.
OR Manually: Mix 200g of the flour, 3 tablespoons sugar, 1 1/2 teaspoons salt and yeast. Heat water and butter to 45 degrees C. Add water to dry ingredients; beat 2 minutes at medium speed on mixer. Add 60g flour, beat at high speed 2 minutes. Stir in more flour gradually to make a stiff dough. Knead 8-10 minutes. Cover and let rise. Punch down. Cover; let rest 15 minutes.
Divide dough into 12 pieces; shape 3 pieces into smooth balls. Poke a 2.5cm hole in each. Drop bagels into a large frying pan with 2.5cm simmering water 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 baking tray.
Bake at 190 C / Gas 5 for 20 to 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.