Make the flatbreads: Stir together flour, baking powder, and salt in a medium bowl. Stir in oil and 3/4 cup warm water until combined. Turn out dough onto a clean surface and knead until smooth, 5 minutes. Cover with a bowl and let rest 30 minutes.
Meanwhile, make the filling: Steam cauliflower until soft, 8 to 10 minutes, then transfer to a medium bowl and mash. Heat oil in a medium skillet over medium heat and toast cumin until fragrant, about 30 seconds, before stirring in coriander and turmeric. Stir in spinach until just wilted. Remove mixture from heat and stir in cauliflower and peas. Season with salt.
Assemble flatbreads: Divide dough into six balls and roll out into 8-inch rounds. Heat a cast-iron skillet over medium heat and cook rounds until slightly puffed and golden, 1 to 2 minutes. Flip and spread half of each round with filling. Gently fold over other half and cook until golden, 2 minutes. Flip and cook 2 minutes more. Brush with oil. Serve flatbreads cut in half.