Stir together 1 Tablespoon kosher salt and 4 cups filtered or distilled water until the salt is completely dissolved. Set aside.
Place 1 crushed garlic clove and 4 sprigs of dill in the bottoms of 2 quart sized jars. Pack the carrot sticks into the jars as tightly as you can. I find the best way to do this is to pack them vertically in 2 layers, making sure to leave some space at the top of the jar for the brine. The tighter you pack them in, the better, as it will help keep them in place when they are fermenting.
Pour the brine into the jars. You may have some leftover, which could be used to make more pickles. If any of your carrots are floating up in the brine, use something to keep them submerged. It could be something edible, like a cabbage leaf, or a small upturned glass container. I use glass weights that are specially designed for fermenting.
Cover the top of the jar with a few layers of cheesecloth, a few coffee filters or a clean cotton dish towel secured tightly with an elastic band. Move to a dark space where they won't be disturbed (like a pantry). Check on them daily to make sure no carrots or dill is floating above the brine, which will cause mold problems. If floating does occur, use a weight to push everything down.
After a few days, you may see the brine begin to get cloudy. This is a normal part of the pickling process. Some pickles will get cloudier than others due to water content. After about a week, start tasting the pickles. They will begin to taste sour. The longer you leave them, the more sour they will become. When they have reached a flavor that you like, remove the weight (if used) and replace the cloth on top with the lid, and place in the refrigerator. They will keep in there for about a month.