Preheat a large skillet on medium high. Add a tablespoon of the oil.
In a small bowl. Mix together the brown sugar, garlic powder, oregano, thyme, allspice, nutmeg, cinnamon, salt, and cayenne together. Rub the mixture onto the shrimp to evenly coat.
Place the shrimp on the hot skillet in an even layer. Cook on both sides for about 3 minutes, or until it just turns opaque. Do not overcook or the shrimp will come out rubbery.
Remove from heat and place the shrimp on a heated plate.
In a large bowl, mix together the remaining oil, lime juice, and apricot preserves then mix well. Add salt and pepper to taste.
Quickly chop the romaine. Toss the romaine, cilantro, carrots, mango, and avocado into the bowl with the dressing and mix to combine the dressing. Top with the shrimp and serve.