Place the chicken breast halves into a saucepan and fill with enough water to cover. Bring to a boil and cook until the chicken is cooked through, about 20 minutes. Remove from water and cool.

Preheat the oven to 375°F (190°C). Remove chicken meat from breasts, discarding the skin and bones. Set aside.

Heat olive oil in a skillet over medium heat. Add green chilies, jalapeno, and garlic and stir for a few minutes until fragrant. Stir in cream cheese and half the Monterey Jack cheese, gradually adding water as the cheese melts. Chop chicken meat and stir into skillet. Remove from heat.

Spoon chicken mixture into tortillas and roll up. Place rolls seam-side down in a 9x13 inch baking dish. Sprinkle remaining Monterey Jack cheese over the top, then pour heavy cream over all.