1. Prepare the dough by whisking together mayonnaise, oil, salt, warm milk, and warm water in a mixing bowl. Sprinkle yeast and sugar over the mixture and let it sit covered for 5 minutes until foamy. Gradually add sifted flour, mixing until a soft dough forms. Cover and allow to rise at room temperature for 1½ to 2 hours until doubled or nearly tripled in size.

2. For the filling, heat olive oil in a skillet over medium heat and cook the cubed chicken until fully cooked, about 2-3 minutes. Remove chicken and set aside. In the same skillet, melt 1 tablespoon butter, add mushrooms, and sauté until moisture evaporates. Add another tablespoon butter and onions, cooking until tender. Add remaining butter with shredded cabbage, grated carrots, minced garlic, water, salt, and pepper. Cook until cabbage is at desired tenderness. Stir in ketchup, sour cream, fresh herbs, paprika, and cooked chicken. Remove from heat and cool completely.

3. On a lightly floured surface, divide the dough into two equal parts. Roll each into a roughly 12-inch round. Place half the cooled filling in the center of each round, leaving about 3 inches of dough bare at the edges. Fold over the edges, pinching seams tightly to seal. Flip the pirog so the seam is underneath and flatten gently. Repeat for the second pirog.

4. Place pirogi on a greased or parchment-lined baking sheet. Brush tops with egg wash (beaten egg and water). Sprinkle with sesame seeds if desired. Let rise for 30 to 45 minutes until puffy. Preheat oven to 400°F (200°C) and bake for 18 to 22 minutes or until golden brown. Serve warm.