Preheat the oven to 350°F (175°C). Spray a 13x9-inch baking dish with nonstick spray and set aside.
Bring a large pot of salted water to a boil and cook the manicotti shells according to package directions until almost al dente; drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion, minced garlic, and minced jalapeños and cook until softened, about 4–6 minutes.
Remove the skillet from heat and stir in the shredded cooked chicken, room-temperature cream cheese, and sour cream until fully combined and smooth. Let the mixture cool for about 20 minutes, then stir in 1/3 cup of the salsa.
Use a spoon to stuff each manicotti shell with the chicken mixture and place the stuffed shells in the prepared baking dish.
In a small bowl, combine the remaining 1 cup of salsa with the canned enchilada sauce and mix well. Pour 1/2 cup of this sauce mixture into the bottom of the baking dish, then arrange the stuffed shells on top.
If any chicken filling remains, spoon it over the shells. Pour the remaining sauce over the manicotti so they are evenly coated.
Mix the Monterey Jack, sharp cheddar, and Parmesan cheeses together and sprinkle evenly over the top.
Bake for 35–40 minutes, until the sauce is bubbly and the cheese is golden. Let rest a few minutes before serving.
