1. Preheat broiler.
2. Cut chile in half lengthwise; discard seeds and membranes. Place chile halves, skin sides up, on a foil-lined baking sheet; flatten with hand. Broil 6 minutes or until blackened. Place in a paper bag; fold to close tightly. Let stand 5 minutes. Peel and chop chile.
3. Preheat oven to 350°.
4. Combine chile, uncooked quinoa, masa harina, 1 cup water, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a medium bowl; stir until a soft dough forms. Divide mixture into 4 equal pieces; shape each piece into a 4-inch patty. Heat a large ovenproof skillet over medium-high heat. Add canola oil to pan; swirl to coat. Add sopes to pan; cook 3 minutes or until browned. Turn sopes over; place pan in oven. Bake at 350° for 10 minutes or until browned and heated through.
5. Combine remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, olive oil, juice, and sugar in a medium bowl, stirring with a whisk. Add arugula and remaining ingredients; toss gently to coat. Serve over sopes.