Melt butter. Add butter to a saucepot and melt it over medium-low heat. Once the butter is melted, stir in olive oil.
Peel and slice the potatoes. Preheat oven to 450ºF. Peel the potatoes. Cut the potatoes into ½-inch thick slices crosswise. Add the potatoes and thyme to the butter mixture. Toss well to coat the potatoes. Season the potatoes with salt and pepper.
Bake the potatoes. Arrange the potatoes on a rimmed baking sheet in a single layer. Drizzle any remaining butter over the potatoes. Bake the potatoes for 15 minutes. Flip them gently and bake for another 15 minutes.
Add the broth and bake again. Combine vegetable broth with garlic. Pour the vegetable broth mixture over the potatoes. Bake the potatoes for another 15 minutes.
Serve. Drizzle any cooking liquid over the melting potatoes and serve them warm.