Preheat the oven and brush the inside of a 9 by 13-inch casserole dish with oil.
Heat the oil in a large skillet over medium heat. Add the onion and cook until softened, about 5 minutes, stirring occasionally. Add the kale, cover the skillet, and cook until wilted, about 1 to 2 minutes. Stir in the garlic and cook 1 minute more, stirring constantly.
Add the oregano, cumin, salt, black pepper, cream cheese, milk, and 1/2 cup salsa verde. Bring up to a simmer, then add the chicken and simmer 1 to 2 minutes, adding more milk to thin the sauce out if needed. Turn off the heat and stir in 1/3 of the shredded cheddar cheese.
Measure 1/2 cup salsa verde and reserve it for topping; spread the rest of the salsa verde in the bottom of the oiled dish.
Wrap a slightly scant 1/2 cup of the chicken filling in a tortilla and place them seam-side-down in the prepared dish (you should get 8 enchiladas). Spoon the reserved 1/2 cup salsa verde on top, and sprinkle on the reserved shredded cheddar cheese.
Bake uncovered until the enchiladas start to brown and are warm throughout, about 10 to 15 minutes.
Sprinkle the scallion on top and serve warm.