Slice the fingerling potatoes into rounds and add to a medium saucepan. Cover them with 1 inch salted water and bring to a boil. Cook until the potatoes are fork-tender, about 8 to 10 minutes. Peel and mince the shallot. Thinly slice the celery. Finely chop the dill fronds. Drain and rinse the kidney beans.

In a large bowl, whisk together just 1 packet Dijon mustard, just 1 tbsp sweet pickle relish, apple cider vinegar, and 1 tbsp olive oil. Add half the minced shallot, sliced celery, chopped dill, kidney beans, and a pinch of salt and pepper. Drain the potatoes and transfer to the large bowl. Toss the Dijon potato salad well and chill until you’re ready to serve.

Place a large nonstick skillet over high heat with 2 inches water and bring to a boil. Scrub, rinse, and trim the carrots. Add the carrots to the boiling water and cook until fork-tender, about 10 to 12 minutes.

Drain the water from the skillet and return the cooked carrots to medium heat. Add 1 tbsp vegetable oil and the barbecue seasoning. Cook, tossing occasionally, until the carrots begin to char in places, about 4 to 6 minutes.

Set the oven to broil on high. Slice the buns halfway through. Toast the hot dog buns until the tops are toasted, about 1 to 2 minutes.

Open hot dog buns and nestle a carrot dog into each. Top with the remaining Dijon mustard, remaining shallot, sauerkraut, and remaining sweet pickle relish. Serve with the Dijon potato salad on the side.