Preheat oven to 425°F (220°C).
Brush eggplant slices with olive oil and season with smoked paprika, garlic powder, cumin, salt, and pepper. Roast for 20-25 minutes, flipping halfway through, until tender and golden.
While eggplant is roasting, make caramelized onions. Heat olive oil in a pan over medium-low heat. Add sliced onions and cook slowly, stirring occasionally, for about 25-30 minutes until golden brown. Add balsamic vinegar and brown sugar, cook for another 5 minutes. Set aside.
Mix all aioli ingredients in a small bowl. Refrigerate until ready to use.
Prepare the flax egg by mixing ground flaxseed with water and let it sit for 5 minutes to thicken.
Once eggplant is roasted, let cool slightly. Mash it in a bowl and mix with panko breadcrumbs and flax egg.
Form mixture into 4 patties. Heat a skillet over medium heat with a bit of oil. Cook patties for 3-4 minutes on each side until golden and crispy.
Toast vegan burger buns. Spread vegan smoky aioli on both halves.
Place eggplant patty on bottom bun, top with caramelized onions, tomato slice, and arugula. Close with top bun.