Peel the carrots and cut them into sticks. Pack them into a half-gallon jar.

Add the slices of jalapeño pepper to the jar, slipping them in between the carrots and along the side.

Dissolve the salt and sugar in the water. (If you heat the water to make it easier to dissolve the salt, allow it to cool to room temperature before proceeding.)

Pour the brine over the vegetable in the jar. Place a glass weight on top of the vegetables, if necessary, to keep in submerged. Cover with the lid and insert in the airlock into the lid. Set aside out of direct sunlight in a room between 65 and 75°F.

Check the carrots daily to make sure they remain submerged in the brine. You will see small bubbles begin to form and collect on the surface of the brine. If there is any scum (white, yellow, or bluish growth), carefully scoop it off without stirring it into the liquid, and discard.

When it is no longer bubbling, the fermentation is complete. This process can take anywhere from 1 to 2 weeks, depending on the temperature of the environment. Once the carrots are to your liking, store them in the refrigerator, dividing them into smaller jars if desired.