For the aioli: Stir together the mayonnaise, adobo sauce and lime juice in a small bowl. Stir in enough water (a tablespoon or so) to make a thick sauce that you can drizzle. Refrigerate while you make the taquitos to blend the flavors.
For the taquitos: Heat the olive oil in a medium nonstick skillet over medium heat. Add the sweet potatoes and onions and season with salt. Cook, tossing occasionally, until the onion softens, about 8 minutes. Add the garlic and cumin and toss. Make a space in the center of the pan and add the adobo sauce and tomato paste; let toast for a minute and then stir into the vegetables. Add 1/2 cup water and bring to a simmer. Cook until the sweet potatoes are very tender and the sauce is very thick, about 5 minutes (you can add a few more tablespoons of water if the pan dries out too quickly). Stir in the lime zest and transfer to a medium bowl to cool slightly, about 10 minutes.
Heat 1 inch of vegetable oil in a straight-sided skillet until it reaches 350 degrees F on a deep-fry thermometer. Wrap the tortillas in a damp kitchen towel and microwave until warmed and pliable, about 1 minute.
Working with a couple tortillas at a time, form about 2 tablespoons of filling into a log and roll in a tortilla to make a cigar shape, leaving the ends open. Secure closed with toothpicks. Repeat with the remaining tortillas and filling.
Fry the taquitos in 2 or 3 batches (don't crowd the pan) until crisp and golden, 3 to 4 minutes per batch, letting the oil return to 350 degrees F between batches. Drain on a paper-towel-lined baking sheet and season with salt. Remove the toothpicks.
Serve the taquitos on a bed of romaine, topped with the kimchi, a drizzle of aioli and the cilantro leaves.