To prepare the carrots: In a medium-size bowl, whisk together all the ingredients, except the carrots, until the sugar and salt dissolve. Add the carrots, mixing to coat well, and allow to sit for at least 20 minutes or as long as overnight. Drain the liquid before serving.
To make the burgers: preheat a grill, grill pan, or broiler. Place all the ingredients in a medium- size bowl and, using your hands, a silicone spatula, or a wooden spoon, mix to combine well.
Portion the burgers—you can divide the meat into four quarter-pound burgers or make the grown-up burgers slightly larger and the kid burgers slightly smaller—and cook over medium-high heat until cooked all the way through, 5 to 7 minutes per side, though the exact cooking time will vary depending on thickness. Serve on a roll with the quick-pickled carrots on top and mayonnaise mixed with sriracha to taste.