Roast the poblano chilis by rubbing them with oil, then broiling them on a sheet pan. Watch and turn until all sides are fully blackened and charred. Place the chiles in a bowl and cover for 15 minutes, then peel, seed, and chop.
Rinse the tomatillos of their sticky outer coating, then place them in a medium saucepan with the serrano chiles, half of the onion and one cup of water. Cook until softened, then transfer to a blender along with the poblanos, cilantro, crema, garlic, cumin, and salt. Blend on high until the sauce is completely smooth.
Heat vegetable oil in saute pan over medium heat, and fry the tortillas one at a time until pliable. Finely mince the other half of the onion and place it in a bowl along with the shredded chicken or turkey, and one cup of the sauce, stirring to combine.
Cover the bottom of the baking dish in the sauce. Divide the meat mixture among the tortillas, then roll each up like a cigar, placing them seam-side down in the pan. Pour the remaining sauce on top, then cover with cheese.
Bake the enchiladas until the cheese has melted and the sauce is bubbling. Serve warm.