Prepare the Skillet and Potatoes: Preheat your oven to 375°F (190°C). Brush the inside of a 10-inch cast-iron skillet with olive oil and set aside. Peel the sweet potatoes and Yukon gold potatoes, then use a mandoline or vegetable slicer to cut them into very thin slices.
Toss the Potatoes: In a large bowl, gently toss the potato slices with salt, pepper, garlic powder, thyme, sage, parsley, rosemary, grated cheese (reserving a little for the top), and chopped onions. Mix carefully to avoid breaking the delicate potato slices.
Assemble the Gratin: Working with small handfuls of potatoes, arrange the coated slices in the prepared skillet or a 2-quart baking dish, layering them evenly. Sprinkle the reserved cheese over the top. Pour the milk and chicken broth evenly over the potatoes.
Bake the Gratin: Cover the skillet with foil and seal the edges. Bake for 50 minutes. Remove the foil and bake for another 15-20 minutes, or until the potatoes are tender and the top is golden. To check tenderness, pierce the potatoes with a knife.
Serve: Let the gratin cool slightly before serving hot. The cheese will be bubbly, and the herbs will give the dish a fragrant, savory touch.