Properly measure the all purpose flour by spooning the flour in to a measuring cup. This ensures that you aren't over packing the cup. Add flour to a large bowl along with the instant yeast, salt, and sugar. Whisk to combine.
Add the unsweetened plant milk, container of yogurt, and vegan butter in to a small pot. Whisk until the butter is melted and the temperature of the mixture is approximately 125F-130F.
Pour the wet mixture in to the dry and stir until combined with a wooden spoon. When the mixture is crumbly, bring it together with your hands. You might need 1-2 tbsps more of plant milk. Turn the dough on to a clean surface and knead in to a smooth ball.
Place the dough in a lightly oiled bowl and cover with plastic wrap. Place in a warm place for 1 hour.
Lightly flour a clean surface and divide the dough in to 6 even pieces. Roll each piece in to a ball and flatten with your hands.
Add a decent amount of feta and mozzarella cheese in the center and fold all the sides over to close the cheese in the dough. Flip over and roll out the dough. You want it to roughly be 6-7in in length and about ½in thick.
Heat a large cast iron skillet on medium-high heat. Melt a few tablespoon of vegan butter and brush on both sides of the naan bread. When the pan is really hot, cook the naan bread for about 2 minutes or until the bread bubbles and turns golden brown. Flip and cook for another 1-2 minutes. Continue until all the naan bread has been cooked. Turn down heat as needed.
Serve immediately so the cheese stays warm and melty!