Combine first 4 ingredients in a medium bowl, stirring until sugar dissolves. Place onion in a small saucepan; cover with water. Bring to a boil; drain and plunge onion in ice water. Drain onion; add to juice mixture. Chill until ready to serve.
Heat a large cast-iron skillet over high heat. Sprinkle chicken with oregano, cumin, salt, and pepper; toss to coat. Coat pan with cooking spray. Add chicken to pan; cook 4 minutes or until browned and done, stirring occasionally.
Heat tortillas according to package directions. Divide chicken evenly among tortillas. Drain onion; divide evenly among tortillas. Top each tortilla with 2 tablespoons avocado and 1 tablespoon cheese; fold over. Serve with lime wedges, if desired.