Place quinoa in a fine-mesh sieve and rinse thoroughly with cold water. In a small saucepan, bring 3/4 cup water to a boil over high heat. Add a pinch of salt, then stir in quinoa. Cover, reduce heat to low and cook until water has absorbed and quinoa is tender, 12 to 14 minutes. (Tilt pan to make sure water has absorbed; do not stir.) Turn off heat and let quinoa steam, covered, for 5 minutes. Fluff with a fork.
In a small skillet, warm olive oil over medium heat. Add onion and a pinch of salt; cook, stirring occasionally, until onion is soft and translucent, about 5 minutes.
Place quinoa, onion, pinto beans, cilantro, walnuts, cumin, 1/2 tsp. salt and paprika in a food processor; pulse to combine. Transfer mixture to a bowl. With slightly moistened hands, form mixture into 4 3/4-inch-thick patties.
Prepare grill for direct-heat cooking over medium-hot charcoal (medium heat for gas). Preheat grill grate for 10 minutes, then lightly oil it with canola oil. Brush patties all over with olive oil and place on grill; cover if using a gas grill. Grill, carefully turning once, until golden brown on both sides, about 8 minutes total. For the last minute of cooking, place a slice of cheese on each burger and let melt. Serve burgers on warmed corn tortillas with green chilies and other condiments, if desired.