Place all of the above ingredients in a 2-quart saucepan except the ham steaks, and bring to a boil. Reduce heat to medium, and continue to cook until the root beer is reduced by 2/3 and takes on a syrupy consistency, about 20 minutes. Remove the glaze from the heat and strain into a clean, heatproof container.
Heat a large, 12-inch nonstick saute pan over medium heat until hot. Saute the ham steaks 1 at a time for 3 minutes on each side. To serve, place each ham steak on a large plate with the sweet potato puree on the side and drizzle everything with the root beer glaze.
Sweet Potato Puree: Place potatoes in a medium saucepan and add enough cold water to cover by 2 inches. Bring to a boil over high heat. Reduce the heat so that the potatoes remain at a low boil and cook until fork tender, about 10 minutes. Drain in a colander placed in the sink.
Transfer the potatoes to the bowl of a food processor along with all remaining ingredients and process until fairly smooth, 1 to 2 minutes. Return the potato puree to the saucepan and warm gently over low heat, stirring frequently, before serving.