Set a strainer over a bowl and pour salsa into strainer. Strain out excess liquid until 1 ½ cups salsa remains. Discard liquid.
Heat oil over medium-low heat in a large nonstick skillet. Place 2 tortillas in the warm oil, heat for about 1 minute, then turn and repeat on the other side. Place warmed tortillas in a single layer on a work surface, and repeat with remaining tortillas. Wipe out the skillet with clean paper towels, and place skillet over low heat.
Place about 2 tablespoons shredded chicken across the center of each tortilla. Add about 1 tablespoon chopped onion, and 1 tablespoon cheese on top of chicken. Reserve extra cheese for later use.
Roll up each enchilada; place in the skillet, seam side down.
Pour salsa over enchiladas, and sprinkle with remaining cheese. Cover and heat enchiladas gently until cheese is melted, 3 to 5 minutes.
Top with dollops or sour cream, and garnish with fresh cilantro sprigs and lime wedges.
