Mix the egg, water, oil, sugar and yeast together until the yeast dissolves. In a separate bowl, mix the two types of flour with the salt. Add 2/3 of the flour mix to the liquid mixture and stir together. Continue adding the last part of the flour mixture until you can touch the dough without it sticking to your fingers. Knead the dough for about 5 minutes, then add a small cube of cold butter. Knead until it is incorporated into the dough.
Let the dough rise under a cloth for one hour. Place the dough on a board sprinkled with flour. Divide the dough into 10 equal portions. Gently roll into equal-sized bun shapes and place on a baking sheet. Let rise for another half hour. Brush with egg and sprinkle with sesame seeds.
Bake the buns at 400 degrees for about 12-15 minutes. They must have a bit of color and sound hollow when you knock them on the bottom.
Red Onion Relish
Cut onions into thin strips and fry at low to medium heat in butter until lightly caramelized. It will take approx. 10-15 minutes. Add thyme, brown sugar, red wine and vinegar. Let cook until the liquid has boiled down. Season with salt and pepper.
Honey Mustard Dressing
Stir the mayonnaise, honey and mustard. Season with lemon juice, salt and pepper to taste.
Additional ingredients (for six burgers)
Form 6 equal patties of meat. Season generously with salt and pepper. Put on a scorching grill for 3-4 minutes on each side - however, the needed time depends slightly on the heat of the grill and desired doneness. After flipping the burgers, add cheese on the grilled side. Warm the burger buns on the grill off of direct heat.
Split the buns and spread dressing on both cut sides. Add a leaf of lettuce and a slice of tomato. Top with the burger and relish. Enjoy!