Pour the latke mix into a shallow dish. Dredge the thin chicken breasts in the mix on both sides, making sure to completely cover the chicken.
Heat the 1/4 cup oil in a large skillet over medium heat. Pan-fry the chicken breasts in the oil until browned, about 5 minutes on each side. (Add additional oil, if necessary, halfway through.) Place the chicken on a paper towel-lined baking sheet in a single layer and keep warm in a 200 degree oven.
Discard any residue from the bottom of the pan and old cooking oil. Add the remaining 2 tablespoons of oil. Add the shallots and sweat them for 2 minutes until translucent. Add the mushrooms and cook for 10 minutes, or until they've reduced in size and are golden.
Increase the heat to medium-high. Add the apple cider and de-glaze the pan. Cook for 3-5 minutes, or until liquid has reduced and thickened.
Sprinkle in the 2 tablespoons of flour and allow to cook for 1 minute. Whisk in the chicken broth slowly and add the chopped rosemary. Increase the heat to bring the sauce to a bubble.
Reduce the heat to low and allow the sauce to simmer for 5 minutes and thicken. Season to taste AFTER the sauce is at the right consistency. (If you season it before it reduces and thickens, it could get too salty.)
Arrange the chicken on a serving plate. Pour the sauce over the chicken and garnish with additional whole sprigs of rosemary. Serve immediately.