Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
Shred the peeled carrots using a box grater or the fine shredding disc of a food processor; set aside.
Place the drained chickpeas in a bowl and mash with a fork until mostly broken down but still a bit chunky.
Add the shredded carrots, vegan mayo, Dijon mustard, 2 tsp ume plum vinegar, celery seeds, dulse flakes, and minced dill to the mashed chickpeas; stir until combined.
Taste and season with kosher salt and more ume plum vinegar as needed until the mixture tastes balanced and slightly tangy.
Butter one side of each slice of sourdough (or brush with olive oil). Place four slices butter-side down on the prepared sheet and top each with a slice of vegan cheese and a scoop of the chickpea-carrot mixture.
Top each sandwich with the remaining bread slices, butter-side up, and bake on the sheet for about 10–12 minutes, until the bread is toasty and the cheese begins to melt.
Remove from the oven, let cool 1–2 minutes, then slice and serve warm.
