Preheat oven to 400 degrees F. In a small bowl, combine the carrots, jalapeños, rice vinegar, and sugar. Toss to combine and let sit for about 10 minutes. Drain, then set aside.
In another small bowl, mix the mayonnaise, tomato paste, hot sauce, garlic, salt, and pepper. Mix well and set aside.
Patty up the beef into 4 large burgers. You could go as thick or thin as you want, but I just quartered my ground beef and made a patty out of each one. It made for a decent size patty. Season each patty with curry powder, salt, and pepper. It's pretty simple seasoning here, but I think the curry powder really adds a depth of flavor.
In a large skillet, heat the oil under medium-high heat. Add the patties and cook until nicely browned on each side. Flip only once, took about 15 minutes or so, depending on how thick your patties are.
While the patties are cooking, you can start crisping up your ciabatta rolls. Cut each roll in half and spread butter on cut sides of bread. Set bread, cut side up, and bake for about 5 minutes so you get a nice, lightly toasted roll.
Spread spicy mayo on on side of roll, then place patty on roll, and top with pickled carrots and jalapeños.