Preheat oven to 400 degrees F (200 degrees C). Grease a baking sheet lightly.
Mix feta cheese, sour cream, cottage cheese, cream cheese, and eggs together in a bowl. Whisk milk and oil together in a small bowl.
Brush milk-oil mixture over 3 sheets of phyllo dough and stack together. Place some of the cheese mixture in a straight line on one end of the dough. Roll up tightly to form a spiral shape. Transfer spiral to the baking sheet. Repeat with remaining milk-oil mixture, phyllo dough, and cheese mixture.
Bake burek in the preheated oven until golden brown, about 40 minutes.
