Whisk whole-wheat flour, all-purpose flour, baking powder and salt in a medium bowl.
Make a well in the center of the dry ingredients, pour in water and 2 tablespoons oil and whisk just until smooth. Stir in caramelized onions.
Heat a large nonstick skillet over medium-high heat. Lightly brush the pan with some of the remaining 1 tablespoon oil.
Using about 1/4 cup batter for each, pour the batter for 2 flatbreads into the pan and spread each into an approximately 5-inch circle using the back of a spoon. Sprinkle with a pinch of flaky salt (if using) and cook until the bread looks bubbly and mostly dry, 1 1/2 to 2 minutes. Flip and cook until golden on the second side, 1 to 2 minutes more.
Remove from the skillet and wrap in foil to keep warm.
Reduce the heat to medium, brush the pan with more oil and cook 2 more flatbreads, reducing the heat as necessary to prevent over browning. Repeat in 3 more batches with the remaining oil and batter.