Using a small-hole strainer, drain liquid from salsa; discard liquid.
Transfer the salsa chunk mixture to a medium bowl.
Prep any additional toppings as needed (ex: dice onions, dice peppers, wash spinach, etc).
Add cheese and any additional toppings to the salsa mixture and mix together until combined.
Divide cheese mixture evenly over half of each tortilla (about 1/2 cup each).
Fold tortillas in half.
Heat large griddle or skillet over medium-high heat.
Drizzle oil in the skillet.
Place filled tortillas on the skillet (as space allows).
Cook, carefully flipping once, until tortillas are golden brown and crisp and cheese filling melts, about 5 minutes.
Cut quesadillas into wedges and enjoy!