Preheat the oven to 400°F. Cut each potato widthwise into about 4-6 sections, each about 1-2 inches wide. Use a cookie cutter about 2-inches wide to cut each section into a potato round. Save the edges for another use. Dry the potatoes very well using a paper towel.
In an oven-safe skillet, heat the butter and olive oil over medium-high heat. Add the potato rounds and cook them for about 5-7 minutes on each side, or until they are nicely browned and crispy.
Turn off the heat. Season with salt and pepper, and add the garlic and fresh rosemary. Carefully pour the vegetable broth into the skillet. Spoon some of the broth over the potatoes.
Transfer the skillet to the preheated oven and let the potatoes braise for about 20-25 minutes, or until they are fork tender and cooked through. The broth will reduce and create a thick sauce.
Spoon the sauce over the potatoes and serve immediately.