Preheat the oven to 450°F (230°C).
Place the poblano peppers on a baking sheet and roast in the oven for about 15 minutes, turning occasionally, until the skins are blistered and charred.
Transfer the peppers to a bowl and cover with plastic wrap. Let them steam for 10 minutes to loosen the skins.
Carefully peel off the skins of the peppers, being careful not to tear them. Make a slit down the side of each pepper and remove the seeds and membranes.
In a skillet, heat 1 tablespoon vegetable oil over medium heat. Add the chopped onion and minced garlic and sauté until translucent, about 3 minutes.
Add the shredded chicken, ground cumin, dried oregano, salt, and pepper to the skillet. Stir and cook for 3-4 minutes until heated through and well combined. Remove from heat.
Stuff each poblano pepper with the chicken mixture, then sprinkle shredded Monterey Jack cheese inside each pepper.
In a clean mixing 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 egg yolks into the beaten egg whites until combined, being careful not to deflate the mixture.
Spread the flour on a plate. Lightly dust each stuffed pepper with flour, shaking off excess.
Heat about 1 inch of vegetable oil in a large skillet over medium heat until hot but not smoking.
Carefully dip each stuffed pepper into the egg batter, coating completely.
Fry the peppers in the hot oil for about 3-4 minutes per side, or until the batter is golden brown and cooked through. Remove and drain on paper towels.
To prepare the tomato sauce, heat 1 tablespoon vegetable oil in a small saucepan over medium heat.
Add the chopped onion and garlic and sauté until softened, about 3 minutes.
Add the chopped tomatoes, salt, and pepper. Cook, stirring occasionally, for 10-12 minutes until the tomatoes break down and the sauce thickens.
Use an immersion blender or transfer the sauce to a blender and puree until smooth.
Serve the fried chicken chiles rellenos hot, topped with the fresh tomato sauce.