Preheat your oven to 350°F (175°C). Prepare a 9x13-inch baking dish by spraying it thoroughly with nonstick cooking spray to prevent the enchiladas from sticking during baking.

In a medium-sized bowl, combine the 2 cups of shredded, cooked chicken with 1 cup of shredded Monterey Jack cheese. Season the mixture with salt, pepper, and Adobo seasoning to taste, ensuring the filling is well-seasoned for maximum flavor.

Divide the chicken and cheese mixture evenly among the 10 flour tortillas. Place a portion of filling in the center of each tortilla, then roll them up tightly. Arrange the filled tortillas seam-side down in your prepared baking dish, placing them snugly next to each other.

In a medium saucepan over medium heat, melt 3 tablespoons of butter. Once melted, whisk in 3 tablespoons of all-purpose flour to create a roux. Cook this mixture for 1 minute, whisking constantly to prevent burning. This roux will help thicken your sauce.

Slowly pour in 2 cups of chicken broth while continuously whisking to avoid lumps, creating a smooth sauce base. Once well incorporated, stir in 1 cup of sour cream and the entire 4-ounce can of diced green chiles with their liquid. Keep the heat medium-low and do not allow the mixture to boil, as this could cause the sour cream to separate.

Remove the sauce from heat once it's smooth and creamy. Pour it evenly over the rolled tortillas in the baking dish, making sure to cover all the enchiladas with the sauce for maximum flavor and moisture.

Sprinkle 1 cup of shredded Monterey Jack cheese evenly over the sauce-covered enchiladas. The cheese will melt during baking, creating a delicious golden top layer.

Place the baking dish in your preheated oven and bake for 20-25 minutes, until the enchiladas are heated through and the cheese is melted. For an extra golden top, turn the broiler on high for 1-2 minutes at the end of cooking time, watching carefully to prevent burning.