Preheat oven to 350°F (175°C).

Heat olive oil in a large skillet over medium-high heat.

Add lamb shoulder to the skillet and cook until browned on all sides, stirring occasionally.

Add onions and continue cooking until they are translucent and the lamb is cooked through.

Season with salt, pepper, and dried thyme.

In a separate oven-safe dish, layer half of the sliced potatoes.

Pour half of the lamb mixture over the potatoes.

Repeat the layering process with the remaining potatoes and lamb mixture.

Pour beef broth over the top of the dish.

Cover the dish with foil and bake in the preheated oven for 1 hour.

Remove foil and continue baking for an additional 30 minutes, or until the potatoes are tender and the top is golden brown.

Place meat and potatoes on buns.

Serve hot and enjoy!