In a small bowl or cup, whisk together the warm milk, yeast, and sugar. Allow the mixture to sit for 10 minutes, until it becomes frothy.
While the warm soy milk mixture sits, stir the room temperature milk and vinegar together in a large mixing bowl.
Pour the yeast mixture into the bowl with the vinegar mixture. Stir in the oil and salt.
Stir in 1 cup of all-purpose flour and 1 cup of whole wheat pastry flour, to form a soft dough. Continue adding up to an additional 1/2 cup of all-purpose flour, a bit at a time, until the dough is firm enough for kneading (it will still be very soft).
Turn the dough onto a lightly floured surface and knead it until smooth and elastic, about 8 minutes, adding up to 1/2 cup additional flour as needed.
Rub the dough with a bit of oil and place in a clean bowl. Cover it with a damp towel and allow it to rise in a warm location until doubled in size, about 1 hour.
Use a knife to cut the dough into 4 equal portions, then use a rolling pin to roll each into an oval shape, about 10 inches by 4 inches.
Melt about 2 tablespoons of butter in a medium nonstick skillet over medium-high heat. Place one of your dough ovals into the skillet and cover it.
Allow the dough to cook for 2 minutes. Take a peek, and if the dough has gotten puffy and formed brown spots on the bottom, flip it. Cover and cook it about 2 minutes more, until brown spots form on the other side.
Remove the naan from the skillet and transfer it to a plate to cool slightly.
Repeat for the remaining dough portions, adding vegan butter to the skillet as needed between loaves.
Serve your naan warm with additional butter and optional cilantro.
Make sure your skillet is fully heated before adding the dough. I've been finding that the first loaf I make isn't as fluffy as the others, and I suspect this is because I'm impatient and adding the dough too soon. Give it a few minutes.