In a large saucepan, bring the chicken and other deshebrada ingredients to a boil. Reduce the heat to a low simmer and cook the chicken until cooked through and very tender, 25 to 30 minutes. Let the chicken cool for a few minutes in the liquid. Remove the breasts to a plate and let them sit for a few more minutes, until cool enough to handle. Shred the chicken into bite-size pieces. Save the cooking liquid for soups or sauces. (The chicken can be prepared ahead to this point and refrigerated for 1 or 2 days, or frozen for several months.)

Preheat the oven to 350°F. Grease a large baking dish.

In a shallow bowl or dish, combine the sauce with the crema.

Spread about ½ cup of the sauce mixture thinly in the baking dish.

Heat ½ to 1 inch of oil in a small skillet until the oil ripples. With tongs, dunk a tortilla in the oil long enough for it to go limp, a matter of seconds. Don’t let the tortilla turn crisp. Repeat with the remaining tortillas and drain them.

Dip a tortilla into the sauce. Top it with about 1/3 cup of chicken, a couple of teaspoons of onion, and about a tablespoon of cheese. Roll up the tortilla snug but not tight. Transfer the enchilada to the baking dish. Repeat with the rest of the tortillas and filling. Top the enchiladas with any remaining onion and pour the sauce evenly over them. Scatter the rest of the cheese over the sauce.

Bake the enchiladas for 15 to 18 minutes, until they are heated through and the sauce is bubbly. With a spatula, serve the enchiladas immediately.

Regional Variations: In Mexico, these are enchiladas suisas, or Swiss, because they call for crema. Central Texans might substitute sour cream instead. Farther west in El Paso and north into New Mexico, both would be nixed in favor of Monterey jack or cheddar cheese, the enchiladas might be flat rather than rolled, and the area’s green chiles would flavor the sauce, rather than share the billing with tomatillos. Out in California, their former home, Bon Appétit columnists Jinx and Jefferson Morgan made a version with jalapeños, sour cream, and spinach, a rendition so beloved that they proposed it as a substitute for turkey at Thanksgiving.