Place cubed sweet potatoes in a pot with cold salted water. Bring to a boil, then reduce heat and simmer until very tender, 12-15 minutes. Drain thoroughly and let steam in the colander for 5 minutes to remove excess moisture. This step is crucial - wet potatoes make soggy burgers.

Transfer sweet potatoes to a large bowl and mash until mostly smooth with a potato masher. Add chickpeas and mash roughly - you want some texture remaining, not a complete puree. Some whole chickpea pieces are fine and help with the burger's structure.

Stir in panko, smoked paprika, cumin, garlic powder, salt, and pepper. Let cool for 5 minutes, then mix in the beaten egg until just combined. The mixture should hold together when squeezed but still feel slightly moist. If too wet, add 2 more tablespoons panko.

Divide mixture into 4 equal portions and form into 1-inch thick patties, about 3.5 inches across. Press firmly so they hold together. Place on a plate and refrigerate for 10 minutes while you heat the pan - this firms them up significantly.

Heat 1.5 tablespoons olive oil in a large nonstick or cast iron skillet over medium heat. When oil shimmers, add patties and cook undisturbed until deep golden brown and crisp on the bottom, 4-5 minutes. Carefully flip, add remaining oil around the edges, and cook another 4-5 minutes until the second side is equally crisp and the patties feel firm to the touch.

Serve immediately on toasted buns with sliced avocado and arugula. A smear of chipotle mayo or tahini sauce works beautifully if you want something saucy.