In a bowl, whisk the flours with the salt and garam masala. Add the butter and rub the flours between your fingers until the mixture resembles fine cornmeal. Add the jalapeños, onion, cilantro, water and the 1 tablespoon of oil and mix with your hands to form a soft, slightly sticky dough; transfer to a clean bowl, cover with plastic wrap and let rest for 20 minutes.
Break off a golf-ball-size piece of dough. On a lightly floured work surface, roll the dough into a tight ball, then flatten it into a 5-inch disk. Using a paring knife, make a cut from the center of the disk to the edge. Starting with one side of the slit, roll the dough onto itself to form a cone. Flatten it into a disk. On the lightly floured work surface, roll out the disk until 1/8 inch thick; transfer to a lightly floured baking sheet and cover with plastic wrap. Repeat with the remaining dough.
Heat a griddle. Add the flatbreads in batches and cook over moderately high heat until bubbles form on the bottom, about 1 minute. Turn the flatbreads and cook until bubbles form again. Spoon a small amount of oil over the flatbreads. Using the back of a spoon, rub the oil into the flatbreads and cook, pressing, until evenly browned on the bottom, about 1 minute. Flip the flatbreads and repeat with more oil; serve hot.