In a large mixing bowl, add the drained and rinsed chickpeas. Using a fork or potato masher, gently mash the chickpeas, leaving some chunks for texture.
To the mashed chickpeas, add the shredded carrots, vegan mayo, Dijon mustard, ume plum vinegar, celery seeds, dulse seaweed flakes, minced dill, and kosher salt. Mix everything together until well combined. Taste and adjust the seasoning with more ume plum vinegar or salt, if desired.
Lay out 4 slices of sourdough bread on a clean surface. Generously spread the chickpea mixture over each slice. Top with your desired amount of vegan cheese, then place another slice of bread on top to form a sandwich.
Heat a skillet over medium heat and add a tablespoon of vegan butter or olive oil. Once hot, place the assembled sandwiches in the skillet. Toast for about 4-5 minutes on each side, or until the bread is golden brown and the cheese is melted.
Remove the sandwiches from the skillet and let them cool for a minute before slicing in half. Serve warm, and enjoy your delicious Vegan Carrot Chickpea Tuna Melt!