Make the special sauce: Stir all the sauce ingredients together in a small bowl. Season to taste with salt and pepper and set aside.
Make the sliders: Mix the ground beef, garlic and onion powders, salt, Worcestershire sauce and pepper in a large bowl. Form the meat mixture into ten 2-inch patties.
Heat a large cast-iron skillet over medium-high heat. Add enough canola oil to coat the pan and heat until the oil is smoking. Add the patties and cook, flipping once, until heavily caramelized on both sides, 4 to 6 minutes for medium rare. Top each patty with a slice of cheese. Transfer the patties to a plate. Wipe out the pan, add the butter and melt over low heat. Add the buns cut-side down and toast them in the butter.
Spread the chipotle special sauce on both cut sides of each bun. Place a patty and a piece of lettuce on each bun. Top each slider with a pickled jalapeno strip and a pickled carrot slice, securing them with a bamboo pick.