In the first step of making burger buns, pour warm milk and water into a bowl, add sugar and yeast, mix until smooth, cover with a kitchen towel and leave in a warm place to rise for about 15 minutes.
At the second stage, pour flour into a large bowl, add salt and swollen mass. Knead the dough for 4-5 minutes, if kneading by hand, if using a mixer, then 2-3 minutes will be enough. Then add the butter in 2-3 doses and continue kneading the dough. As a result, you should get a soft dough that does not stick to your hands. Pour in some olive oil, cover with a kitchen towel and let rise in a warm place for 1 hour.
Transfer the puff pastry to a floured surface, shape it into a round shape and divide into 8 equal portions. Give each part a round shape, put on a baking sheet, on which you first spread baking paper. The dough sheets should be spaced 3-4 cm apart because they will inflate and not stick to each other.
Brush dough balls with beaten egg in 2 doses, the first time before rising, and the second time just before baking. Cover the dough balls with a kitchen towel and let them rise until they double in size.
Lubricate the balloons a second time with a beaten egg, sprinkle with sesame seeds and bake in an oven preheated to 190 degrees (put about 1 cup of water under the form in another form in the oven to create steam in the oven, this will make the loaves more airy and plump), bake 15 minutes, after this time, remove the form with water and continue to bake burger buns for another 10 minutes.
To prepare the sauce, combine all ingredients and mix well.
Fry cutlets for 4-5 minutes on each side (you can fry both on the grill and in a grill pan or in a regular pan), when turning over the cutlets, after about 2 minutes, put the cheese so that it melts and makes the meat even tastier.
Cut the hamburger buns in half, butter each side and brown the buns on the grill/pan, this will take about 40-60 seconds.
To assemble the burgers, spread sauce on the inside of both slices of bread, top with a patty, a thin slice of tomato, pickle slices, and a lettuce iceberg, top with the other slice of bread and patty. Ready.