In a medium skillet, warm olive oil over medium heat until shimmering. Add carrots and ginger. Season lightly with salt and pepper and add 2 Tbsp. water. Cover and cook, stirring occasionally, until tender, about 7 minutes. Uncover, raise heat to high and cook, stirring frequently, until carrots have browned in spots, 3 to 5 minutes more.
Transfer to a food processor and let cool slightly. Add lime juice and 1 tsp. sesame oil and pulse until coarsely combined. Add Hearts of Palm Dip and pulse until smooth. Taste and season with salt and pepper, if desired. Transfer to a serving bowl. Drizzle with remaining 1/2 tsp. sesame oil, sprinkle with sesame seeds and serve.