I should note that there is no bun, this is just the meat patty she's eating. It seems less weird if you picture them as ground sausage patties instead.
It's definitely a good idea nutritionally: protein is the best thing to have first thing in the morning, as it is easier to digest than carbs, and burns more slowly so your blood sugar doesn't spike after a long night of no intake. And as far as meats go, ground meats are easiest to digest, plus I mix a ton of pureed zucchini in when I make them, so she's getting secret veggie nutrients too.