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Rinse and trim the green beans into 2-inch lengths. Slice the ham steaks into ribbons approximately the same size as the green beans. Reserve.
Whisk the chicken broth, soy sauce, mirin or balsamic vinegar, and hot sesame oil together in a small bowl. Whisk in the cornstarch.
Heat the vegetable oil in a wok or large skillet over high heat. (Do not use a nonstick skillet to stir-fry; the high heat destroys the finish of the pan.) Add the ham and cook, stirring constantly, until brown.
Add the green beans and cook for 2 minutes, stirring constantly, then add the garlic and cook, still stirring, for a minute more.
Stir in the soy broth and the cornstarch slurry and cook until the sauce thickens, about 2 minutes.
Serve with rice or other grain or noodle of your choice.