Bring a pot of water to a rolling boil. Add the carrot sticks and garlic cloves, blanching them for 1 to 2 minutes. Remove from the water and allow them to cool while preparing the jars.

To sterilize the jars, submerge them along with their lids in simmering water for 10 minutes or place them in a dishwasher cycle just before use. Keep the jars hot until ready to fill.

Pack the blanched carrot sticks and garlic cloves snugly into the sterilized jars, ensuring an even distribution.

In a medium saucepan, combine the water, cider vinegar, granulated sugar, light brown sugar, and kosher salt. Stir well and bring the mixture to a boil. Add the mustard seeds and whole peppercorns, then let the brine simmer for 5 minutes, allowing the flavors to meld. Strain the mixture through a fine strainer to remove the seeds and peppercorns for a smoother finish.

While the vinegar mixture is still hot, carefully ladle it into the jars, filling them until the liquid reaches ½ inch from the top. Wipe the jar rims with a clean, damp cloth to ensure a proper seal. Secure the lids and rings tightly.

Process the sealed jars in a boiling water bath, ensuring the water level is at least 1 inch above the tops of the jars. Let them boil for 10 minutes. Once finished, remove the jars from the water bath and place them on a countertop to cool completely.

These pickled carrots are ready to enjoy after they have had time to fully develop their flavors. Serve them as a tangy addition to an antipasto platter or toss them with marinated artichokes, cherry tomatoes, black olives, fresh mozzarella, and chopped Romaine for a vibrant antipasto salad dressed with a simple Italian vinaigrette.