Cut the chicken breast halves lengthwise into 1/2-inch-wide strips.
In a medium bowl, combine the tomato paste, chili powder, garlic powder, and 1 Tbs. of the olive oil. Add the chicken strips and stir to coat. Marinate for 10 minutes.
Meanwhile, combine the mango, avocado, red pepper, cilantro, and the juice of 1 lime in a medium bowl. Season with salt and pepper to taste. Cut the remaining lime into wedges.
Heat the remaining 2 Tbs. olive oil in a 10-inch sauté pan over medium- high heat. Season the chicken strips with salt and pepper. Sauté the strips, stirring often, until they’re firm to the touch and uniformly opaque, 3 to 4 minutes. Turn off the heat and let sit for 1 minute.
Set the chicken strips on a platter, top with the salsa, fan the lime wedges around, and serve.