Preheat oven to 375 degrees. Scoop out cores of apples with a melon baller. Use the same tool to remove blossom scars at bottoms of apple halves and any skin remaining at stem ends.
Place sugar in a 10-inch ovenproof skillet over medium heat. When sugar begins to melt, stir with a wooden spoon. It will form big lumps at first, but keep cooking and stirring until all the sugar is dissolved and forms a light-amber syrup. As soon as it reaches this stage, remove from heat, and add butter to prevent caramel from overcooking. Stir until butter is melted and incorporated. Arrange apples cut side up on top of caramel, and tuck rosemary sprigs between them. Place skillet in oven, and bake for 15 minutes. Using tongs, turn apples over; bake until apples are softened, about 15 minutes more.
Using a serrated knife, cut shortcakes in half horizontally. Place the bottom half of each shortcake cut side up on a large dessert plate. With a small spatula, lift the hot caramelized apple halves from syrup, and place each one cut side down on a shortcake bottom. Remove and discard rosemary. Return skillet with caramel to stovetop, and bring it to a simmer over medium heat. Gradually whisk in cream. When sauce is smooth, remove from heat.
Top each apple half with a scoop of ice cream. Drizzle the caramel sauce over the top of each dessert, letting it drip onto the plate. Place the shortcake tops over each. Serve immediately.