Heat the oven to 375°F. Lightly grease a 9x13-inch baking dish.
Make the creamy sauce: In a medium saucepan over medium-low heat, whisk the enchilada sauce and cream cheese until completely smooth and melted, 3 to 4 minutes. Remove from heat and whisk in the sour cream, cumin, chili powder, garlic powder, jalapeños, and lime juice. Fold in the black beans.
In a small bowl, combine the cheddar and Monterey Jack cheeses and set aside.
In a large bowl, toss the shredded chicken with 1 cup of the warm sauce to coat. Reserve the remaining sauce for layering.
Warm the tortillas in a dry skillet over medium heat, 10 to 15 seconds per side, until pliable. Tip: Stack them in a clean kitchen towel as you go—keeping them warm prevents cracking and helps them absorb just the right amount of sauce.
Spread about 1/3 cup of the reserved sauce over the bottom of the prepared dish. Arrange a layer of tortillas (3 to 4, tearing to fit) over the sauce.
Scatter half of the sauced chicken over the tortillas. Sprinkle with about one-third of the cheese mixture, then drizzle with 1/2 cup of the reserved sauce.
Repeat with another layer of tortillas, the remaining sauced chicken, another third of the cheese, and 1/2 cup sauce. Top with a final layer of tortillas, spread the remaining sauce all the way to the edges, and finish with the remaining cheese.
Cover the dish tightly with foil and bake for 22 minutes. Uncover and bake 8 minutes more, then switch the oven to broil and broil on high for 2 minutes, until the cheese is bubbly and lightly blistered.
Let the casserole rest for 10 minutes before slicing so the layers set. Serve warm.