Heat broiler with the rack positioned 4-5 inches from the element, and line two baking sheets with foil or parchment.
Brush both sides of eggplant slices with olive oil and sprinkle liberally with salt.
Arrange slices on prepared baking sheets and broil - one sheet at a time - until eggplant slices are a deep mahogany brown, turning once halfway through, 5 to 7 minutes per side.
Adjust the oven to 375°F, and position rack in the center.
In a 10-inch skillet, heat 1 tablespoon olive oil over medium heat.
Add onion and sauté until translucent, but not browned, stirring occasionally, about 5 minutes.
Add ground lamb, stirring frequently, breaking up meat into very small pieces with the side of a wooden spoon.
Season with salt, cinnamon, and pepper.
Sauté until meat is just cooked through, then taste for seasoning and adjust accordingly.
In a small skillet, melt butter over medium heat. Add pine nuts, stir to coat with the butter, and reduce heat to medium-low.
Cook, stirring frequently, until nuts are golden brown, 2 to 4 minutes. Keep a close watch over the nuts; they can burn quickly once they begin to brown.
Transfer nuts to a bowl while still warm and salt them lightly.
Brush a medium oval baking dish - approximately 6-inch by 10-inches - with olive oil.
Spread 1/4 cup of tomato sauce in the bottom of the dish.
Lay 1/3 of the eggplant slices in a single layer over the sauce, covering as much surface area of the bottom of the dish as possible.
Spoon 1/2 the meat evenly over eggplant.
Pour 1/3 of the remaining tomato sauce evenly over meat.
Sprinkle with 1/3 of the pine nuts. Layer again with eggplant, meat, tomato sauce and pine nuts.
Finish with a third layer of eggplant and cover with the remaining tomato sauce, sprinkling remaining pine nuts on top.
Pour warm chicken stock around the perimeter of the baking dish. (Sauce will thicken as it bakes.)
Cover pan with foil and bake for 90 minutes. Remove foil and top eggplant evenly with mozzarella.
Start cooking the rice; bake casserole uncovered for approximately 20 minutes, or until the cheese is bubbling and golden.
It should be done about the same time as the rice.
Serve warm in deep bowls over fluffed rice.