Make the Cooked Green Salsa:
Put the tomatillos and jalapeños in a medium saucepan, pour in enough cold water to barely cover (about 3½ cups), and bring to a boil. Cook until the jalapeños are soft and the tomatillos are tender, about 20 minutes. Remove from the heat and let stand for 15 minutes to finish cooking the inside of the tomatillos.
Drain the tomatillos and chiles (gently, to aviod breaking up the tomatillos) in a colander. Wipe out the saucepan and set aside. Put the tomatillos, jalapeñnos, garlic, and cumin in a blender jar and blend for a few seconds, just until the tomatillos are coarsely chopped. Add the cilantro (see Notes) and blend until the sauce is smooth and speckled with finely chopped cilantro. Do not overblend, or you will grind the tomatillo seeds and make a thick and pasty rather than smooth and shiny sauce.
Heat the oil in the cleaned pan over medium heat. Pour in the salsa, bring to a simmer, and simmer until lightly thickened, about 10 minutes. Season with the salt. The sauce can be stored in the refrigerator for up to 3 days. Reheat over low heat before using.
Preheat the oven to 375°F.
Make the enchilada sauce:
Pour half the green sauce into a blender jar, add the poblanos, and blend at low speed until smooth. Pour into a medium saucepan, add the remaining green sauce and the cream, and bring to a simmer over low heat. Remove from the heat. If prepared in advance, reheat, adding water 1 tablespoon at a time if necessary to return the sauce to the right consistency, before ladling over the enchiladas.
If using store-bought tortillas, soften in the oil and drain them. Fill and roll the tortillas, using 1/3 cup of the chicken for each one. Place the enchiladas seam side down in a 9 by 13-inch baking dish. The enchiladas can be prepared to this point up to 4 hours in advance. Cover them with a damp kitchen towel or paper towels, cover the dish with plastic wrap, and refrigerate. Remove from the refrigerator 30 minutes before baking.
Pour the hot sauce over the enchiladas. Jiggle the dish so the sauce settles in between the enchiladas. Sprinkle the cheese evenly over the top. Bake until the sauce around the edges is bubbling and the cheese is golden brown, about 20 minutes. Let stand for 5 minutes before serving.
Scatter the onion and cilantro over the enchiladas. Scoop 2 enchiladas and plenty of sauce onto each serving plate.
VARIATION: Green Salsa For the Table.
Coarsely grind 2 roasted and peeled jalapeños with a large pinch of salt in a molcajete or in a mortar with a pestle. Stir in 2 cups cooled Cooked Green Salsa, ½ cup finely chopped white onion, ½ cup chopped fresh cilantro, and salt to taste. (Alternatively, blend the chopped jalapeños and sauce in a blender jar using quick on/off pulses just until the jalapeños are coarsely chopped. Scrape into a bowl and stir in the onion, cilantro, and salt.) Let stand for 1 hour before serving.