Whisk egg whites and eggs together in a large bowl.
Heat 1 teaspoon oil in a large frying pan over medium-high heat. Cook and stir hash browns until browned and crispy, about 5 minutes. Transfer hash browns to a large bowl.
Heat 1 teaspoon oil in the same frying pan over medium heat. Add turkey, onion, and ham. Cook and stir until turkey is crumbly and onion softens, about 5 minutes. Add eggs; stir until scrambled and set, 5 to 7 minutes. Add cheese and let melt, about 3 minutes. Pour mixture into the bowl with the hash browns. Mix everything together.
Plate tortillas. Add about 3 tablespoons of filling to each tortilla. Fold opposing edges of a tortilla to overlap the filling. Roll 1 of the opposing edges around the filling, creating a burrito. Repeat for remaining tortillas.
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