Preheat the broiler or grill to high heat.
Place the poblano peppers on a baking sheet and roast under the broiler or on the grill, turning occasionally, until the skins are charred and blistered on all sides, about 10 minutes.
Transfer the roasted peppers to a bowl and cover with plastic wrap. Let them steam for 15 minutes to loosen the skins.
Carefully peel the skins off the peppers, then make a slit down one side of each pepper and remove the seeds and membranes, keeping the peppers intact.
In a medium skillet, heat 1 tablespoon vegetable oil over medium heat. Add 1/2 cup chopped onion and 2 minced garlic cloves; sauté until softened, about 3 minutes.
Add the shredded chicken, ground cumin, dried oregano, 1/4 teaspoon salt, and 1/4 teaspoon black pepper to the skillet. Stir to combine and cook for 2 minutes until heated through. Remove from heat and set aside.
Stuff each poblano pepper with the seasoned shredded chicken mixture, being careful not to tear the peppers.
Separate the egg whites and yolks into two large bowls. Add cream of tartar to the egg whites and beat with an electric mixer until stiff peaks form.
In the bowl with the egg yolks, add 1/2 teaspoon salt and beat until smooth. Gently fold the yolks into the beaten egg whites to create a fluffy batter.
Lightly dust each stuffed pepper with all-purpose flour, shaking off any excess.
Heat vegetable oil in a large skillet over medium-high heat, about 1/2 inch deep for frying.
Dip each floured stuffed pepper into the egg batter, coating completely, then carefully place into the hot oil.
Fry the peppers until golden brown on all sides, about 3-4 minutes per side. Remove and drain on paper towels.
To prepare the sauce, heat 1 tablespoon vegetable oil in a saucepan over medium heat. Add 1/4 cup chopped onion and 1 minced garlic clove; sauté until translucent, about 3 minutes.
Add the crushed tomatoes, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Simmer the sauce for 10 minutes, stirring occasionally.
Serve the fried chicken chiles rellenos hot, topped with the warm tomato sauce.