Preheat oven to 425. Line a large baking sheet with parchment paper.
Scrub and dry the potatoes. Slice into matchsticks. Toss with 2 tablespoons olive oil, 2 tsp turmeric, chili powder, garlic powder, salt, and pepper. Roast for 35-40 minutes until crispy, tossing halfway through.
In the bowl of a food processor, blitz the carrots until smooth (not pureed, just no longer chunky). Add the chickpeas, flour, parsley, lemon juice, garlic, turmeric, cumin, coriander, red onion, and a pinch of salt and pepper. Puree until well combined. The dough should be cohesive and fairly dry, but not crumbly.
Form the dough into 12-16 1-inch balls and set on a sheet of parchment. Press lightly to flatten into discs. Prepare a large plate with a layer of paper towels.
Heat 1/4 cup olive oil in a large skillet over medium heat until shimmering. Place the falafels into the oil and cook 3 minutes per side, until crispy and golden. You may need to do this in batches - drizzle a bit more oil to the pan if necessary between batches. Set the falafels on the paper towel-lined plate to drain.
To assemble: place a tuft of fresh arugula on one side of a naan loaf. Arrange cucumbers and pickled red onions overtop, then divide the falafels among the loaves. Sprinkle with parsley and drizzle with feta sauce. Serve with golden fries, additional veggies and sauce on the side.