Mix flour, 1 teaspoon salt, and yeast in a large bowl. Stir in olive oil and about 3/4 cup water until mixture comes together in a ball of dough. Turn dough out on a lightly floured surface. Gradually add remaining 1/4 cup water while kneading dough; continue kneading until smooth and elastic, about 10 minutes.
Place dough in a large lightly-oiled bowl and turn to coat. Cover with a towel and let rise in a warm place , about 30 minutes.
Melt butter in a large skillet over medium heat. Cook and stir onions and 1 teaspoon salt until tender and translucent, about 30 minutes; stir often so onions do not brown. Remove from heat and allow to cool.
Preheat oven to 375 degrees F (190 degrees C). Grease a baking sheet.
Roll dough out on a floured surface to create a thin crust; transfer dough to prepared baking sheet. Poke several holes in the dough with a fork and roll edges up slightly to create a rim.
Spread cooled onions over crust. Whisk sour cream, milk, eggs, and ground black pepper in a bowl; pour over onions. Sprinkle with caraway seeds.
Bake in preheated oven until golden brown and eggs are set, 30 to 45 minutes. Serve warm.