For the feta, make the pesto by combining the mint, parsley, garlic, and olive oil in a blender container. Blend until smooth. Add the nuts and pepper; blend. Marinate the feta slices in the pesto for several hours or overnight.
For the hummus, in a medium saucepan, combine the olive oil and garlic cloves. Cook until lightly toasted, stirring occasionally. Stir in the cayenne and cumin. Add the chickpeas. Simmer for 5 minutes. Transfer to the bowl of a food processor. Puree with the lemon juice and enough water to make a thick puree. Season to taste with salt and white pepper.
For the carrots, blanch the carrots in a large pot of boiling water until crisp-tender, about 2 to 3 minutes. Drain; toss with honey. Place in a shallow roasting pan. Season to taste with salt and pepper. Set aside.
For the lamb, heat the oven to 425° F. Place a medium roasting pan in the oven for 5 minutes to heat. Add 1 tablespoon olive oil. Season the lamb all over with salt and pepper. Place the meat in the hot roasting pan. Roast 10 minutes. Add 3 garlic cloves and the thyme sprigs. Turn the lamb over. Roast an additional 10 to 15 minutes to 130° F for medium-rare. Remove from the oven. Let stand, covered with foil, for 5 minutes. Roast the carrots during the last 10 minutes of cooking time.
To serve, place a mound of hummus in the center of each plate. Slice the lamb and place 2 or 3 chops on top of the hummus. Place the feta and carrots on top, and drizzle a bit of the pesto around the plate