Lyonnaise potatoes are traditionally made with regular potatoes, but work just as well with sweet potatoes. If you have some on hand, try adding toasted chopped pecans or a dollop of Greek yogurt.
Heat the oil in a large skillet or cast-iron pan over medium-high heat. Once the oil is hot, add the onions. Season with salt and pepper and cook the onions, stirring often, until softened and browned around the edges, about 5 minutes. Transfer the onions to a bowl.
While the onions are cooking, peel and cut the sweet potatoes into 1/4-inch slices.
Reduce the heat to medium and add the butter to the skillet, swirling to prevent burning. When the butter has melted, cover the bottom of the skillet with as many potato slices that will fit in one layer (this should be about half of the potatoes). Cook until the potatoes are deeply browned on one side, about 4 minutes. Remove to the bowl with the onions. Add the remaining potato slices, and then spread the cooked onions/potatoes over top. Season with salt and pepper, pour in the chicken broth and cover the skillet.
When the potatoes are partly tender with some resistance, about 3 minutes, use a metal spatula to flip the bottom layer to the top and continue cooking. (Don’t worry; it will take 3 to 4 flips to transfer all the browned potatoes to the top.) Cook until the potatoes are tender, about 2 more minutes. Remove the lid, raising the heat to medium-high and allow the remaining liquid to evaporate, about 3 minutes.