Prepare the cheese filling: In a bowl, mix together the crumbled feta, shredded gouda (or spicy cheddar), provolone, beaten egg, and salt & pepper.
Shape the dough: Roll the pizza dough into two 10" ovals, leaving a 1" border around the edges.
Assemble the khachapuri: Place the cheese mixture in the center of each dough oval, then fold the edges over the cheese to form a boat shape. Optional: Add thin lemon slices on top for added flavor.
Add the herb topping: In a small bowl, mix the chopped parsley, cilantro, dill, and grated garlic. Sprinkle the herb mixture evenly over the cheese before baking.
Prepare for baking: Brush the edges of the dough with a beaten egg and sprinkle with sesame seeds.
Bake: Place the boats on a baking sheet and bake in a preheated oven at 450°F for 20 minutes, until the crust is golden and the cheese is bubbly.
Finish baking: Remove from the oven, crack an egg yolk into the center of each boat, and bake for an additional 3-5 minutes, or until the egg yolk is just set.
Make the hot honey butter: In a small saucepan, melt together the butter, honey, smoked paprika, harissa (optional), chili flakes, and a pinch of salt. Stir until the mixture is warm and well combined.
Serve: Drizzle the hot honey butter over the baked khachapuri and serve immediately. Swirl the egg yolk and melted cheese together while hot for a rich, creamy texture.