For fresh tomatoes: Roast the tomatoes and chiles on a baking sheet 4 inches below a very hot broiler, until they're darkly roasted (they'll be blackened in spots), about 6 minutes. Flip them over and roast the other side—5 or 6 minutes more will give you splotchy-black and blistered tomatoes that are cooked through. Cool. Working over your baking sheet, pull off and discard the blackened tomato skins and, for round tomatoes, cut out the hard "cores" where the stems were attached. Transfer tomatoes and chiles to a food processor or blender, along with all the juices on the baking sheet. Blend to a smooth puree.
For canned tomatoes: In a small dry skillet, roast the chiles over medium heat, turning regularly, until they're soft and splotchy-black, about 5 minutes. Place in a blender or food processor along with the drained canned tomatoes. Blend to a smooth puree.
In a medium-size (4- or 5-quart) pot (preferably a Dutch oven or Mexican cazuela), heat the oil or lard over medium heat. Add the onion and cook, stirring regularly, until golden, about 7 minutes. Raise the heat to medium-high, and, when noticeably hotter, stir in the tomato puree. Cook, stirring, until darker in color and thickened to the consistency of tomato paste, about 10 to 15 minutes. Stir in the broth, partially cover and simmer 15 minutes. Taste and season with salt, usually about 1/2 teaspoon. The sauce should be a slightly soupy consistency—not as thick as spaghetti sauce. If it is too thick, stir in a little additional broth. Keep warm over low heat.
Other preliminaries. Stir the crema (or one of its stand-ins) into the sauce. Put the chicken in a bowl and stir 1/2 cup of the sauce mixture into it. Taste and season with additional salt if you think it needs it. Have the cheese at the ready.
Heat the oven to 350°. Smear about 1/4 cup of the sauce over the bottom of 4 to 6 nine-inch individual ovenproof baking/serving dishes or smear about 1 cup of the sauce over the bottom of a 13x9-inch baking dish. Lay the tortillas out on a baking sheet (2 sheets if you have them, for more even heating), and lightly brush or spray both sides of the tortillas with oil. Bake just to warm through and soften, about 3 minutes. Stack the tortillas and cover with a towel to keep warm.
Working quickly so the tortillas stay hot and pliable, roll a portion of the chicken into each tortilla, then line them all up in the baking dishes. Douse evenly with the remaining sauce, then sprinkle with the cheese. Bake until the enchiladas are hot through (the cheese will have begun to brown), about 15 minutes. Garnish with onion rings and cilantro sprigs. These are best served piping hot from the oven.
MAKE AHEAD The sauce can be made a day or two ahead; refrigerate covered. Once the tortillas have been heated in the oven, you need to work quickly and steadily toward serving in order to preserve their beautiful texture. Once out of the oven, the finished dish softens to near mush over a period of 15 to 20 minutes.