Over medium-high heat, melt the butter in a medium saucepan.
Add the flour and cook for 1-2 minutes while stirring.
Whisk in the broth and salsa. Cook over medium high heat until thickened. It should coat the back of a spoon nicely. Salt and pepper to taste. Remove from heat.
In a large bowl, combine the rice, shredded chicken and half of the sauce.
Spread approx 1/4 cup of filling into the first tortilla, top with a little shredded cheese and roll up. Place in a glass 9x13 dish.
Repeat with remaining filling and tortillas.
Once all of the tortillas have been filled and are lined up in the pan, drizzle the remaining sauce over the top of the enchiladas and then sprinkle any remaining cheese on top.
Bake in the oven at 350° until the sauce and cheese are hot and bubbly. Approx 10 minutes.
Serve hot and enjoy!