This is a delicious side dish, especially during the holidays. I use a mandolin to slice everything, but it still takes a bit of prep time. It makes a great side to any pork dish and is a great presentation.
Position rack in bottom third of oven and preheat to 450°F. Brush 9-inch-diameter cake pan with 1 1/2-inch-high sides with 1/2 tablespoon butter. Line bottom of pan with parchment; brush parchment with 1/2 tablespoon butter.
Place apple slices, 1 tablespoon butter and 1 teaspoon minced sage in small bowl. Sprinkle with salt and pepper; toss to coat. Place potatoes, remaining 3 tablespoons butter and 2 teaspoons minced sage in medium bowl. Sprinkle with salt and pepper; toss to coat.
Arrange 1 layer of neatest potato slices in concentric circles in prepared pan, overlapping potatoes in outermost circle (this will become top of galette). Top with second layer of potatoes. Top with apple slices. Cover with remaining potatoes. Pour over any butter from bowls used to toss potatoes and apple.
Bake galette until potatoes are tender when tested with tip of sharp knife and top is beginning to brown, about 1 hour.
Remove galette from oven. Turn out onto 9- or 10-inch tart pan bottom; peel off parchment. (Can be made 2 hours ahead. Let stand at room temperature.)
Preheat broiler. Broil galette until top is golden, about 2 minutes. Transfer galette to platter. Cut into wedges.