In a large mixing bowl, combine flour, sugar, salt, and instant yeast. In a separate bowl, mix warm milk, water, and melted butter, then pour into the dry ingredients.

Knead the dough for about 8-10 minutes until smooth and elastic. Place in a greased bowl, cover with a damp cloth, and let rise until doubled in size, about 1 hour.

Heat olive oil in a skillet over medium heat, add diced onion and sauté until translucent, about 3-4 minutes. Stir in minced garlic, tomato paste, smoked paprika, cumin, and hot sauce, cooking for an additional minute.

Incorporate shredded chicken, then fold in cream cheese and shredded mozzarella until combined. Remove from heat and cool slightly.

After the dough rises, punch it down gently, divide into equal portions, and flatten into discs. Place a spoonful of the filling in the center, fold the edges, and pinch to seal.

Place sealed buns seam-side down on a parchment-lined tray and let rest for 10-15 minutes. Preheat oven to 375°F (190°C).

Brush tops with beaten egg and bake for 18-20 minutes until golden brown.

Remove from oven, allow to cool slightly, and serve.