Put the oil and garlic in a small wok or a deep, heavy-based, 10-inch, straight-sided sauté pan with a lid. Set over medium-high heat and cook, stirring frequently, until the garlic begins to sizzle steadily, about 1 minute. Add the bok choy (the pan will be crowded) and, using tongs, turn it in the oil and garlic, and then season it with the salt. When the tender tops begin to wilt, in about 1 minute, add the sherry (or rice wine) and toss again for about 15 seconds before adding the chicken broth, soy sauce, and sugar. Reduce the heat to medium, cover, and simmer until the bok choy tops are completely wilted and the stalks are crisp-tender, about 5 minutes. Transfer the bok choy to a plate.
Give the cornstarch slurry a stir to recombine and then whisk it into the cooking liquid. Simmer vigorously until the liquid has thickened, about 30 seconds. Remove from the heat and return the bok choy to the pan. Add the prosciutto and toss quickly to coat the bok choy with the broth and to mix in the prosciutto. Season to taste with salt and serve.