Khachapuri (Georgian Cheese Bread)

Ingredients (14 items)

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