Pesto: Grate 1 teaspoon zest from the lime and juice the entire lime for about 2 tablespoons of juice. Combine the zest and juice with the cilantro, green onions, chopped pecans, chopped garlic, sugar, vegetable oil, and extra-virgin olive oil in the bowl of a food processor. Pulse the mixture until well-combined, but still slightly chunky. Season the pesto with salt, and pepper, to taste.
Shrimp: In a bowl, toss the shrimp with 1 tablespoon of the pesto. Heat the oil in a large skillet over medium-high heat. Saute the shrimp until just opaque and pink, about 3 minutes. Put a generous spoonful of pesto into a serving bowl. Add the shrimp and toss to coat. Serve the remaining pesto on the side or freeze for another use.