Combine warm water, sugar, and yeast. Leave 5–10 minutes until foamy. Add yogurt, oil, and egg. Add flour, salt, and baking powder and stir to combine. Knead on a lightly floured surface 7–8 minutes until smooth and slightly tacky. Place in an oiled bowl, cover, and rise 1 to 1.5 hours until doubled.
Divide dough into 6 equal pieces. Flatten one piece into a rough 5-inch oval. Place 3–4 tablespoons of mozzarella in the center. Bring edges up and pinch tightly to seal. Turn seam-side down and gently roll into an oval about 7–8 inches long and 1/4 inch thick. Repeat with remaining pieces.
Heat a cast iron skillet or heavy griddle over high heat for 3–4 minutes until very hot. Do not add oil. The pan must be extremely hot for proper blistering and char.
Place one naan in the dry hot pan. Cook 1.5–2 minutes without moving until bubbles form and the bottom has dark spots. Flip and cook 1–2 minutes more. Both sides should have char spots and the cheese inside should be fully melted. Remove and repeat with remaining naan, letting the pan return to full heat between each.
Mix melted butter with grated garlic, cilantro, and salt. Brush generously over each naan immediately after cooking. Serve warm.