Roast the poblano peppers over an open flame or under a broiler, turning occasionally until the skin is charred and blistered on all sides, about 10 minutes.
Place the roasted peppers in a sealed plastic bag or covered bowl to steam for 10 minutes, then peel off the charred skin carefully. Make a lengthwise slit on each pepper and remove the seeds, keeping the peppers intact.
In a skillet, heat 1 tablespoon vegetable oil over medium heat. Add the chopped onion and minced garlic and sauté until softened, about 3 minutes.
Add the shredded chicken, ground cumin, dried oregano, salt, and pepper to the skillet. Cook for 5 minutes, stirring occasionally, until the mixture is heated through and well combined. Remove from heat.
Stuff each poblano pepper with the chicken mixture and some shredded Monterey Jack cheese, then close the slit gently.
Separate the eggs into two large bowls. Beat the egg whites with cream of tartar until stiff peaks form.
In the other bowl, lightly beat the egg yolks, then gently fold them into the beaten egg whites to form a fluffy batter.
Heat about 1 inch of vegetable oil in a deep skillet over medium heat until hot but not smoking.
Lightly dust each stuffed pepper with flour, shaking off excess.
Dip each floured pepper into the egg batter, coating completely.
Carefully place the battered peppers into the hot oil. Fry for about 3-4 minutes on each side or until golden brown and cooked through. Remove and drain on paper towels.
Warm the tomato sauce in a small saucepan over low heat.
Serve the chicken chiles rellenos topped with warm tomato sauce and garnished with chopped fresh cilantro.