Heat milk and butter together until lukewarm (about 105°F) and the butter is melted. Mix the sugar and salt in off the heat until dissolved.
In a separate bowl, whisk together bread flour, salt, and yeast.
Pour the wet mixture into the dry mixture. Using a wooden spoon, mix until a dough forms.
Transfer onto your work surface and knead until smooth (by hand: about 8-10 minutes, with stand mixer: 6-8 minutes on medium speed). If the mixture is too sticky, knead in a tbsp of flour at a time until you have a tacky but not sticky dough.
Transfer into an oiled bowl. Cover and let proof for 2 hours. In the meantime, prepare the shakshuka sauce and cheese mixture.
Sauté onion and bell pepper in olive oil over medium high heat for 2 minutes.
Add jalapeño and garlic, sauté until fragrant.
Add tomato paste and spices, sauté for another minute.
Add tomatoes and cook until thick. We want a saucy topping as opposed to your normal shakshuka sauce.
Season to taste with salt.
Combine grated mozzarella and feta in a bowl. Incorporate using your hands, then mix in water. Incorporate until well combined and set in the fridge.
Preheat oven to 375°F. You can use either a pizza stone or a baking sheet lined with parchment paper for this.
Punch air out of dough, then divide into 4 even balls, about 150g each.
Roll each ball into an oval that’s about 10 inches long by 6 inches wide. Cover with greased plastic and let proof for 20 minutes.
Spoon ¼ of the cheese mixture onto the dough and evenly spread out until it’s an inch away from the edges all around.
Spoon dollops of desired amount of shakshuka topping.
Pull the dough’s edges up around the cheese to form the crust. Twist the top and bottom ends to form a boat shape. Cover with greased plastic and let proof for 20 minutes.
Brush crust with egg wash and transfer into the oven. Bake for 15 minutes, remove, crack an egg into each Khachapuri and back into the oven for 15 minutes more.
Remove and garnish with parsley and chives.