In a large bowl, combine whole wheat flour, cake flour, all-purpose flour, and salt. Add the 2 tablespoons oil; stir with a fork or rub with your fingers until mixture resembles coarse crumbs. Stir in milk until dough forms a ball. Turn dough out onto a lightly floured surface. Knead a few minutes to make a moderately stiff dough. Shape dough into a ball. Place in a lightly greased bowl, turning once to grease surface of dough. Cover; let rest for 30 minutes.
In a deep fryer or deep saucepan, heat 1-1/2 inches of oil to 350 degree F. Meanwhile, divide dough into 12 pieces. Roll pieces into balls, then roll balls into 4- to 5-inch circles. Keep circles covered with plastic wrap until ready to fry.
Carefully add one dough circle to hot oil. When poori puffs, turn it over and fry until second side is golden (about 1 minute total). Remove poori from oil with a slotted spoon; drain on paper towels. Repeat until all pooris have been fried. Serve immediately. Makes 12 pooris.