Step 1

Make sauce: Preheat broiler with rack on rung closest to heat. Set tomatillos on a foil-lined rimmed baking sheet. Broil, turning once, until blackened, 10 minutes. Purée in a blender. Measure 3/4 cup and set aside. Set oven to 350°.

Step 2

Heat oil in a medium saucepan over medium heat, then add shallot and cook until translucent, 2 to 3 minutes. Add thyme, salt, and pepper and cook until shallot starts to brown, 1 minute. Immediately add wine and reduce until almost gone, 2 minutes. Stir in cream, tomatillo purée, and sugar. Cook over medium-high heat, stirring often, about 5 minutes, then reduce heat and simmer until sauce thickly coats a metal spoon, 3 to 5 more minutes. Set aside.

Step 3

Make enchiladas: Heat 2 large frying pans over medium-high heat. In first pan, pour 1/4 in. oil and heat until shimmering. Working with 1 tortilla at a time, cook in dry pan, turning once, until softened, 10 to 20 seconds, then in oil, turning once, until puffy and softened but not crisp, 10 to 15 seconds. Arrange in a single layer on paper towels.

Step 4

Lay tortillas flat on a work surface. Arrange chicken and queso fresco down the centers. Roll closed and set, seams down, in a 9- by 13-in. baking dish (with no sauce on the bottom). Pour sauce over enchiladas; top with Oaxaca cheese.

Step 5

Bake until cheese melts and sauce is bubbling, 20 minutes.

Step 6

Garnish with queso fresco, cilantro, onion, and a drizzle of crema (from a squeeze bottle, if you have one). Add chorizo if you like and serve with salsa.

Step 7

*Find at Latino markets.