Khachapuri (Georgian Cheese Bread)

Ingredients (14)

• + ½ cup warm milk add to pantry
• + ⅓ cup warm water add to pantry
• ✓ 1 ½ teaspoons white sugar
• + 1 (.25 ounce) package active dry yeast add to pantry
• ✓ 2 teaspoons olive oil
• + 2 ¼ cups all-purpose flour, divided add to pantry
• + 1 ½ teaspoons kosher salt add to pantry
• + 4 ounces Monterey Jack cheese, shredded add to pantry
• + 4 ounces low-moisture mozzarella cheese, shredded add to pantry
• + 8 ounces feta cheese, crumbled add to pantry
• + 2 large eggs add to pantry
• + 1 tablespoon butter, cut into 4 pats add to pantry
• ✓ sea salt to taste
• + 1 pinch cayenne pepper, or to taste add to pantry