Preheat the oven to 400 degrees F (200 degrees C).
Combine lamb, onion, parsley, salt, allspice, and pepper in a large bowl. Roll meat into 2 1/2-inch long pieces and arrange in a baking pan.
Bake in the preheated oven for 15 minutes.
While meat is cooking, heat vegetable oil in a frying pan over medium heat. Add potatoes; fry until lightly golden, about 5 minutes. Set aside on paper towels to drain excess oil. Sprinkle with salt.
Remove lamb from the oven; drain and discard any fat from meat. Leave oven on.
Combine garlic, lemon juice, water, tahini, salt, and pepper in a food processor until smooth. Set tahini sauce aside.
Arrange potatoes on top of lamb in the baking dish and pour tahini sauce on top. Drizzle with olive oil. Return the baking pan to the oven.
Bake in the preheated oven until lamb is no longer pink and an instant-read thermometer inserted into the center reads at least 150 degrees F (65 degrees C), about 30 minutes more.