Preheat the grill or a grill pan over medium heat.
Toss the eggplant slices with oil and season well with salt and pepper. Grill until tender and nicely marked. Then set aside.
Heat a nonstick pan on medium heat. Once hot, add the oil. Brown the sausage patties on each side to an internal temperature of 150 degrees using a thermometer. Then set aside.
Melt the butter in a saucepan on medium heat. Stir in the flour until fully combined. Cook until golden brown. Whisk in the milk until thick and smooth. Add in the nutmeg a little at a time to taste. Stir in the parmesan until fully combined. Remove from the heat. Season with salt and pepper to taste.
Preheat the oven broiler on high.
To build the towers: Layer one slice of eggplant, sausage patty, another slice of eggplant, sausage patty, and another slice of eggplant. Top with bechamel.
Place under the broiler until nicely browned, about 1 to 3 minutes. (Keep a watchful eye to prevent burning)
Serve with bread. Garnish with thyme.