Heat the olive oil in a large skillet over medium-high heat.
Season the chicken breasts evenly with kosher salt and black pepper on both sides.
Add the chicken breasts to the skillet and cook for 5 to 6 minutes on each side, or until golden brown and cooked through. Remove the chicken from the skillet and set aside.
Reduce the heat to medium and add the minced garlic and chopped jalapeños to the skillet. Cook for 1 to 2 minutes, stirring frequently, until fragrant.
Pour in the heavy cream and chicken broth, stirring to combine and scraping up any browned bits from the bottom of the skillet.
Bring the sauce to a gentle simmer and cook for 3 to 4 minutes until slightly thickened.
Stir in the shredded Monterey Jack and sharp cheddar cheeses until melted and the sauce is creamy.
Return the cooked chicken breasts to the skillet, spooning the sauce over the top.
Simmer for an additional 2 minutes to heat the chicken through and allow flavors to meld.
Remove from heat and sprinkle the chopped fresh cilantro over the skillet before serving.