Bring to a boil a separate small saucepan of water in which to blanch the carrots. Peel the carrots and trim to fit the jars. Cut into thin sticks.
When the water comes to a boil, drop in the carrots and cook for 90 seconds. Remove the carrots from the water and run under cold water to stop the cooking. Divide the spices and garlic cloves between the jars and pack in the carrot sticks on top of the spices.
Combine the vinegar, 1 cup of water, and the salt in a small saucepan and bring to a boil.
Pour the boiling brine over the carrots, leaving 1/2 inch/12 mm of headspace. Tap the jars gently to remove any air bubbles. Add more liquid to return the headspace to 1/2 inch/12 mm, if necessary. Wipe the rims, apply the lids and rings, and process for 10 minutes in a boiling water bath.