Preheat the broiler or grill. Place the poblano peppers on a baking sheet and roast, turning occasionally, until the skins are charred and blistered, about 8-10 minutes.
Transfer the roasted peppers to a bowl and cover with plastic wrap. Let them steam for 10 minutes to loosen the skins.
Carefully peel off the charred skin from the peppers. Make a slit down the side of each pepper and remove the seeds and membranes, keeping the peppers whole for stuffing.
Heat 1 tablespoon of vegetable oil in a skillet over medium heat. Add the chopped onion and cook until softened, about 3 minutes.
Add the minced garlic and cook for another 30 seconds until fragrant.
Stir in the shredded chicken, ground cumin, dried oregano, salt, and black pepper. Cook for 2-3 minutes until heated through. Remove from heat and set aside.
Stuff each poblano pepper with the seasoned chicken mixture, being careful not to tear the peppers.
In a clean bowl, beat the egg whites with cream of tartar until stiff peaks form.
In a separate bowl, lightly beat the egg yolks. Gently fold the yolks into the egg whites until combined.
Spread the flour on a plate. Lightly dust each stuffed pepper with flour, shaking off any excess.
Heat the remaining 1 tablespoon of vegetable oil in a large skillet over medium heat.
Dip each floured pepper into the egg batter, coating evenly, then carefully place in the hot oil.
Fry the peppers for about 3-4 minutes on each side until the batter is golden and cooked through. Remove and drain on paper towels.
For the sauce, combine crushed tomatoes, chicken broth, sugar, and salt in a small saucepan. Simmer over medium heat for 10 minutes, stirring occasionally.
Serve the fried chicken chiles rellenos topped with the warm tomato sauce and garnish with chopped fresh cilantro.