Whisk the flour, yeast, sugar, and salt together in a large bowl. Stir in the yogurt, oil, and warm water until a shaggy dough forms and no dry flour remains.

Turn the dough onto a lightly floured surface and knead until the dough is smooth and elastic, about 8 to 10 minutes. Place the dough in a greased bowl, cover with plastic wrap, and let rise in a warm spot until doubled in size, about 1 to 1.5 hours.

Combine the mashed potatoes, cilantro, chili, cumin, amchur, and salt in a medium bowl. Mash with a fork until the spices are evenly distributed and the mixture is cohesive.

Punch down the dough and divide it into 8 equal pieces, rolling each into a smooth ball. Cover with a damp towel to prevent a skin from forming while you work.

Transfer one dough ball to a lightly floured counter. Flatten the ball using your hands or a rolling pin and roll out to a 6-inch disk. Place a generous 1/4 cup of filling in the center.

Bring the edges of the dough up and over the filling, pinching them together firmly to seal the bundle. Flip the dough over so the seam side is down and roll gently into an 8-inch circle—don't press too hard or the filling will burst through the dough.

Heat a large cast-iron skillet over medium-high heat until a drop of water sizzles and evaporates immediately. Place the naan in the dry skillet and cook until the bottom is charred in spots and the top surface bubbles, about 2 minutes.

Flip the bread and cook until the second side is charred and the bread is cooked through, about 2 minutes more. Transfer to a plate, brush with melted butter, and serve warm.