Heat a large saute pan over medium-high heat and add the oil. Season the lamb with salt and pepper. Sear the lamb racks, fleshiest side down first. Let sit and brown about 2 to 3 minutes. Flip and sear the other side. Repeat with the other rack. Transfer lamb racks to a sheet pan.
Combine the thyme and rosemary and press an even amount into each seared lamb rack. Let sit for a few minutes.
Roast the lamb in the oven for 12 to 15 minutes. Remove from the oven and let rest for 5 minutes.
Heat the apricot jam in a small saucepan over low heat. When the jam is melted, turn off the heat. Brush a light coating of jam on each piece of lamb.
For the sauce: Heat a deep saute pan over medium heat and add the oil, lamb trimmings, and rosemary. Cook until the fat is a little cooked down and the trimmings are deep golden brown, 3 to 5 minutes.
Deglaze the pan with the chicken stock. Reduce by two-thirds.
Stir in the demi glace and pomegranate juice concentrate. Simmer until thickened, about 5 minutes. Remove from the heat.
Strain the sauce and set aside. Just before ready to serve, re-warm the sauce and whisk in the butter. Taste for seasoning. Add salt and pepper, if needed.
For the potato cakes: Preheat the oven to 400 degrees F.
Grate the potatoes and blend with the flour, egg, salt, and pepper.
Drizzle the oil into the small cast iron pans and fill halfway with the grated potatoes. Top the potatoes with salt, pepper, and goat cheese. Top with the remaining potatoes.
Bake the potato cakes in the oven until golden brown and cooked through, 20 to 30 minutes.
Turn the pans over to unmold the potato cakes.
For the julienned vegetables: Heat a saute pan over medium heat and add some oil. Saute the julienned vegetables and season with salt and pepper. Cook until the vegetables are just softened, 3 to 5 minutes. Taste a bite and cook for another minute if the vegetables are still hard.
To plate: Slice off 2 chops from each rack and then cut the meat from the rest of the rack into slices. Serve each guest 1 lamb chop on the bone and 3 slices of boneless meat. Serve with a potato cake and sauteed vegetables.