Peel the potatoes and cut them into evenly sized cubes. Peel and finely chop the garlic.
Line a large baking sheet with parchment paper and arrange the potato cubes on it. Drizzle with olive oil, then sprinkle with salt, pepper, and the chopped garlic.
Place the rosemary sprigs among the potatoes, then mix thoroughly to coat everything evenly with the spices.
Bake in a preheated oven at 400°F (200°C) for 35-40 minutes, or until the potatoes are golden brown and crispy.
While the potatoes are roasting, thinly slice the smoked ham. If desired, warm it through in a skillet over medium heat.
Serve the roasted potatoes and smoked ham hot. Garnish with a fresh salad or pickled vegetables.
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