In a large skillet, combine carrot, sugar, salt and vinegar; cook over medium heat, stirring occasionally, until sugar has dissolved, 2 to 3 minutes. Spread out on a plate and refrigerate until chilled.
In a mini food processor, puree hearts of palm with oil and lime juice until smooth.
Open baguette pieces. Spread half of hearts of palm on cut sides (reserve remaining spread). Arrange cucumbers on bottom half, followed by avocado, pickles, jalapeño, onion, carrot and cilantro. Close sandwiches and serve.