The mole sauces I make are fairly thick, a bit thicker than heavy cream (I'll post a couple or three later), so when I'm using a mole for enchiladas I'll remove some to a large saute pan and thin in a bit--to more the consistency of light cream. (Btw, the dark moles work good with brisket point as well as chicken or turkey.)
For cheese I prefer a Mexican melting cheese like Queso Fresco (often found in supermarkets near the chilled tortillas or near the parmesan wedges), and I like Queso Cotija (and/or Queso Anejo) for crumbling on the top.
I warm my meat (chicken or turkey chunks or shreds, or brisket point chunks) in a pot on the stove if I've pulled it out of the fridge. I use a little weak stock or broth plus any juices I've saved (if I have any left) from when I pulled the meat out of the foil after resting. I just use a little of this liquid mixture for moisture and to help transfer heat. I use a low flame and keep the cover on the pot but stir occasionally. I'm looking to get the meat warm-hot, but not cook it.
Meanwhile, I warm the thinned mole in the saute pan on med-low heat (to a bare simmer) and have a 13x9 pan or 8x8 (or whatever) baking pan nearby.
Though tortillas are generally softened in hot oil, I use the flames from an available burner instead, tossing the tortilla directly on the burner and moving it around quickly with tongs, and flipping, to get it pliable and a little toasty. This is pretty quick. I do them all in succession and stack them under a towel to keep them warm.
[Alternatively, rather then semi-frying the tortillas in oil, do this: Lay the tortillas out on a baking sheet or two, and lightly spray or brush both sides of the tortillas with oil. Bake at 350 just to warm and soften them, about 3 minutes. Stack the tortillas and cover with a towel to keep them warm.]
For assembly: You can spread a little sauce in the bottom of your pan, fill the tortillas one at a time with your meat and a little cheese, and lay them in your pan, topping the lot with more sauce and a little more cheese, or, what I do: I take a warm tortilla dip it entirely into your saute pan of sauce, then remove it to the baking pan I'm using. I sprinkle on a little Queso Fresco, lay in a line of drained meat filling and roll it up. Then I repeat till all are rolled. Rolling in the baking pan saves you from messing up your counter or another pan. As the pan fills just lay your tortilla on top of the finished ones and roll it up right there.
Take some of your sauce (not too much) and pour in over the enchiladas. Top with a little more Queso Fresco, and a little crumbled Cotija and/or Anejo. Bake at 350 till heated through and the cheese is starting to brown, about 15-20 min.
Imo, the biggest problem with restaurant Mexican food (and pizza, for that matter) is there's nearly always too much cheese. I love cheese, but you can kill a dish with too much. I recommend not putting too much in with the meat and not smothering the dish with it. You want your wonderful meats and the nuances in the sauce to shine.
One of the sauces I make is similar to Kevin Taylor's, except that I don't use a roux, tomato paste or chili powder. Instead I roast roma tomatoes under the broiler, flipping once till blackened in spots, then peel them over a blender to catch the juices. I re-hydrate chile peppers in boiling water, then puree them with the peeled tomatoes, a splash of orange juice, the chicken stock, and the spices. I then simmer the sauce till a thick as I'd like. If I'm making a suiza-style dish (with crema or sour cream mixed into the sauce) I skip the o.j. (Btw, if you can get creme fraiche (or make your own, it's easy) it works better than sour cream as it won't break when added to the hot sauce.