1. In a bowl, whisk mayonnaise, oil, salt, warm milk, and water. Sprinkle yeast and sugar over the mixture and let foam for 5 minutes.
2. Gradually incorporate sifted flour to form a dough. Cover and let rise at room temperature for 1 ½ to 2 hours until doubled.
3. Cook chicken in oil over medium-high heat until done, remove and set aside.
4. Sauté onion in 1 tablespoon butter until tender. Add remaining butter, cabbage, carrots, garlic, and water; cook until tender.
5. Stir in ketchup, sour cream, herbs, and cooked chicken. Cook for another minute, then remove from heat to cool.
6. Preheat oven to 400°F (200°C). Divide dough into two parts, roll each to a 12-inch round.
7. Place half the filling on each dough round, leaving 3 inches free at the edge. Fold edges over filling, pinch to seal, flip and flatten.
8. Brush each pirog with egg wash, sprinkle sesame seeds. Let rise for 30-45 minutes.
9. Bake on a parchment-lined sheet at 400°F (200°C) for 18-22 minutes until golden brown.