Heat the oven to 400 degrees. Using 2 tablespoons of the oil, lightly brush both sides of each tortilla. Heat a large skillet or griddle over medium-high and, working in batches, cook tortillas until lightly browned and just starting to crisp but still pliable, 45 seconds to 1 minute per side. (This will prevent the tortillas from completely falling apart while cooking.) Set aside.
Heat the remaining 2 tablespoons of oil in a large skillet over medium-high and cook onion, poblano, garlic, tomatoes and juices, and 1 teaspoon salt, stirring and scraping up any browned bits, until most of the liquid has evaporated and tomatoes begin to stick to the skillet, 9 to 11 minutes. Stir in chicken, ½ cup enchilada sauce and ½ teaspoon salt until combined and just warmed through; remove from heat. Taste and season with additional salt, if necessary.
Pour ½ cup enchilada sauce into a 9-by-13-inch baking dish, spreading to cover the bottom. Working one at a time, arrange a scant ¼ cup of chicken down the center of the tortilla, roll to secure the filling and place seam side down in the baking dish. Repeat with remaining tortillas and chicken, making sure they’re nestled right up against each other.
Pour the remaining enchilada sauce over the tortillas, then scatter cheese on top. Bake until the sauce is bubbling and the cheese is just beginning to brown, 15 to 20 minutes. Serve warm topped with pico de gallo, if desired. For leftovers, divide remaining enchiladas into portions of 2 or 3; wrap each portion tightly in plastic wrap or place in a freezer bag, squeeze out the air, seal and freeze for up to 3 months.