Preheat the oven to 450°F (230°C).
Place the poblano peppers on a baking sheet and roast in the oven for 15-20 minutes, turning occasionally, until the skins are blistered and charred.
Remove the peppers from the oven and place them in a bowl. Cover with plastic wrap and let them steam for 10 minutes.
Peel off the charred skins from the peppers, being careful not to tear them. Make a slit down the side of each pepper and carefully remove the seeds and membranes. Set aside.
In a skillet, heat 1 tablespoon vegetable oil over medium heat. Add the finely chopped onion and minced garlic and sauté until translucent, about 3 minutes.
Add the shredded chicken, ground cumin, dried oregano, salt, and black pepper to the skillet. Stir to combine and cook for 5 minutes until heated through. Remove from heat.
Stuff each poblano pepper with the seasoned shredded chicken mixture and set aside.
In a medium saucepan, heat 1 tablespoon vegetable oil over medium heat. Add the chopped onion and minced garlic, sautéing until softened, about 3 minutes.
Add the crushed tomatoes, salt, and black pepper to the saucepan. Simmer the sauce for 15 minutes, stirring occasionally. Remove from heat and keep warm.
Separate the egg whites and yolks into two clean bowls.
Add the cream of tartar to the egg whites and beat with an electric mixer until stiff peaks form.
In a separate bowl, lightly beat the egg yolks.
Gently fold the beaten egg yolks into the egg whites until combined.
Heat 1 cup vegetable oil in a large skillet over medium-high heat.
Lightly dust each stuffed pepper with flour, shaking off any excess.
Dip each floured pepper into the egg batter, coating completely.
Carefully place the battered peppers into the hot oil and fry for 3-4 minutes on each side, or until golden brown and cooked through.
Remove the peppers from the oil and drain on paper towels.
Serve the chicken chiles rellenos hot, topped with the warm tomato sauce.