For Filled Bagel, first mix sugar, flour, salt and yeast in a baking bowl. Add butter, milk and egg white, mix, knead into a soft, smooth dough.
Cover and let rise at room temperature for about 1 hour until doubled.
Stop kneading the dough and form into a roll on a lightly floured surface. Cut into pieces, shape into balls. Make a hole in the center of each ball with a lightly floured forefinger or the handle of a wooden spoon. (To do this, circle the dough rings around the finger or wooden spoon handle until an opening of about 4 cm ø is formed).
Place on a baking sheet lined with parchment paper, cover and let rise for about 15 minutes.
Bring water to a boil, season with salt. Place only a maximum of 3 bagels at a time in the almost boiling salted water for about 12 seconds. Remove with slotted spoon, drain well, return to parchment paper.
Brush bagels with egg yolk. Sprinkle bagels with poppy seeds, sesame seeds, flax seeds, sunflower seeds and/or coarse-grained salt as desired.
Bake for about 15 minutes in the middle of the oven heated to 220 °C. Remove and cool on a rack. Remove, let cool on a rack.
Cut open the bagels. Spread with cream cheese and top with vegetable slices, eggs and cheese. Garnish with parsley if desired and serve the Stuffed Bagel.