Place the shrimp in a medium bowl. Toss with the olive oil, salt and pepper.
Heat a large nonstick pan over medium-high heat. Add the shrimp and cook until just cooked through, about 1 minute per side. Transfer the shrimp to a plate.
Cut the ends off of the cucumber, and cut the cucumber crosswise into 3 pieces. Run the cucumber through the spiralizer to produce noodles. With kitchen shears cut the noodles into slightly smaller pieces.
In a serving bowl, combine the cucumber noodles, shrimp, avocado and hearts of palm. Toss with the dressing. Serve immediately.
In the bowl of a food processor, combine all of the dressing ingredients. Process until well combined, scraping down the sides as necessary.