Preheat the oven to 400°F.
In a small saucepan, combine the quinoa with 2 cups of water and a pinch of salt. Bring to a simmer, reduce heat to low, cover and cook for 12 minutes. Cool.
On a baking sheet, toss the sweet potatoes and 2 whole jalapeños with the oil, salt and pepper. Roast until tender and starting to brown, about 15-20 minutes.
Place the thawed spinach in a clean dish towel and squeeze out as much water as possible. Place in a food processor and blend until smooth. Add the cooked quinoa, eggs, cheddar cheese, onion, carrots, mayo, parsley, cumin, breadcrumbs, salt and pepper and pulse just to combine, being careful not to overprocess.
Shape the mixture into 4 patties. If the mixture is too wet, add more breadcrumbs until it comes together into patties. If the mixture is too dry, add more mayonnaise.
Heat a large skillet or griddle to high heat with a drizzle of olive oil. When hot, sear the burgers for 4-5 minutes per side, until golden and heated through.
Dice the remaining raw jalapeño and add to a blender, along with the roasted jalapeños and remaining aioli ingredients. Purée until smooth. Season to taste.
Serve the burgers on buns with lettuce, tomato, red onions and jalapeño aioli. Add roasted sweet potatoes on the side.