In a large bowl, whisk the mayonnaise, oil, salt, warm milk, and warm water.
Sprinkle the yeast over the top of the mixture.
Sprinkle the sugar over the yeast.
Cover the bowl with a clean kitchen towel. Let the bowl sit at room temperature until the yeast activates and becomes foamy, about 5 minutes.
Gradually add the flour, mixing until well incorporated.
Cover the bowl and set the dough aside to rise at room temperature until the dough doubles and almost triples in size, about 1-1/2-2 hours.
In a large skillet over medium-high heat, heat the oil.
Add the chicken and sauté, stirring as needed, until fully cooked at 165 degrees F internally, about 2-3 minutes per side.
Transfer the chicken from the skillet.
Add 1 tablespoon butter. Add the onion and sauté, stirring frequently, until tender, about 5 minutes.
Add the remaining butter, cabbage, carrots, garlic, and remaining water.
Season the mixture with salt and pepper.
Sauté the cabbage until it reaches desired tenderness, about 3-6 minutes.
Add the ketchup, sour cream, herb mixture, and chicken, cooking and stirring frequently for 1 minute.
Remove the mixture from the heat to cool.
Lightly flour a work surface.
Divide the dough into two equal parts.
Cover one of the dough halves.
Roll the remaining uncovered dough half out, until it forms a 12-inch round.
Place 1/2 of the cabbage filling in the center of the dough round, spreading it evenly and leaving about 3 inches of dough uncovered around the edges.
Gently start pulling the edges of the dough over the filling into the center to close the pirog, continuing to do so until the cabbage is completely covered.
Pinch the seams together.
Gently flip the pirog over. Flatten the pirog until it is even in size and the filling is evenly distributed within the dough. Gently flip the pirog back over.
Repeat the rolling, filling, and folding process with the remaining dough and the remaining filling.
Brush the pirogs with the egg wash.
Sprinkle the pirogs with sesame seeds.
Preheat the oven to 400 degrees F.
Let the dough rise for 30-45 minutes.
Bake the pirogs until golden in color, about 18-22 minutes.
Serve hot.