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.
Once cooled, peel off the charred skins carefully, keeping the peppers whole. Make a small slit down one side of each pepper and remove the seeds and membranes. Set aside.
Heat 2 tablespoons vegetable oil in a skillet over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes.
Add the minced garlic and cook for 1 more minute until fragrant.
Add the shredded chicken, ground cumin, dried oregano, salt, and black pepper to the skillet. Stir well and cook for 3-4 minutes until heated through. Remove from heat and let cool slightly.
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 the cream of tartar to the egg whites and beat with an electric mixer until stiff peaks form.
In the bowl with the yolks, lightly beat until combined.
Gently fold the beaten egg yolks into the egg whites until just combined, being careful not to deflate the mixture.
Spread the flour on a plate. Carefully dredge each stuffed pepper in the flour, shaking off any excess.
Heat about 2 cups of vegetable oil in a deep skillet or frying pan over medium heat until hot but not smoking (about 350°F or 175°C).
Dip each floured pepper into the egg batter, coating evenly.
Carefully place the battered peppers into the hot oil and fry for about 3-4 minutes on each side or until golden brown and puffed.
Remove the peppers from the oil and drain on paper towels.
To prepare the sauce, heat 1 tablespoon vegetable oil in a saucepan over medium heat.
Add the crushed tomatoes, chicken broth, sugar, and salt to taste. Simmer for 10 minutes, stirring occasionally.
Serve the fried chicken chiles rellenos hot, topped with the warm tomato sauce.