Cut off the 8 largest outer leaves from the lettuce and wash and dry them carefully, trying not to tear them. (Cut the leaves at the base, where they connected to the root; the leaves should be about 9 inches long.) Using a vegetable peeler, shave down the center rib of each leaf so it will roll up more easily. Set aside.
Combine the cream cheese, spinach and basil in a food processor and puree until smooth. Season with 1/4 teaspoon kosher salt and a few grinds of black pepper.
Lay two of the romaine leaves out on a clean work surface. Cover the leaves with 2 slices of turkey. Spread 2 tablespoons of the cream cheese mixture over the turkey and line up 1/4 of the pepper strips along one end. Line up 1/4 cup of the shredded carrots just above the peppers. Roll up the wrap tightly, starting from the edge with the peppers and carrots, so the wrap holds together and the peppers and carrots end up in the center of the roll. Place the wrap on a plate seam-side down. Repeat with the remaining ingredients. Cut each wrap in half and serve.
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