Heat 2 tbsp. oil in medium saucepan over medium high heat; add onions. Cook, stirring occasionally, until soft, about 5 minutes. Add tomato sauce, garlic and cinnamon to saucepan; cook until garlic is fragrant, about 1 minute more. Add chicken bouillon mixture, bay leaf and cilantro sprig to pot; bring liquid to boil. Reduce heat to medium low and simmer, stirring occasionally, until sauce thickens, about 25 minutes. Strain sauce, discarding solids; keep warm.
Meanwhile, place poblano peppers over direct fire of stove. Cook, turning occasionally, until skin is completely charred, about 8 minutes. Transfer peppers to zip-top bags until cool. Peel peppers; rinse under cool water. Using small knife, make 2” slice on side of pepper; remove seeds and rinse under cold water. Using paper towels, pat peppers dry inside and out. Place cheese stick inside each pepper, overlapping sides of slit to enclose cheese. Transfer peppers to refrigerator.
Heat 1½” oil in heavy-bottomed pot over medium-high heat until temperature registers 350°F on deep-fry thermometer. In clean bowl, beat egg whites until stiff peaks form. In separate bowl, beat egg yolks until combined. Gently fold yolks into egg whites.
In batches of two, dredge pepper in flour, tapping off excess; dip in egg mixture and transfer to hot oil. Fry peppers until batter is golden brown and crisp and cheese melts, about 6 minutes, flipping peppers with large metal spoon with holes. Transfer to paper towel-lined plates to drain.