Heat a skillet over medium heat and cook the crumbled chorizo, stirring occasionally, until fully browned and no longer pink, about 5–7 minutes.
Transfer the cooked chorizo to paper towels to drain excess oil, leaving any remaining fond in the skillet.
Reduce heat to low and pour the heavy cream into the skillet.
Gradually add the shredded Monterey Jack cheese to the cream, stirring constantly until the cheese is melted and the mixture is smooth, about 2–5 minutes.
Remove the skillet from the heat, stir the drained chorizo back into the cheese mixture until evenly combined.
Sprinkle with chopped cilantro and serve the choriqueso warm with tortilla chips or warm tortillas.