You can brine pickle, which is to say lacto-ferment, any crunchy vegetable. The general rule is that you want about 2 to 3 percent salt by weight of the vegetable. You don't need to be exact, but too little and the vegetables rot, too much and they are too salty.
Cut the carrots into discs the size you would want to eat at a cocktail party. Or leave them whole. Or split them in half lengthwise. The key here is to keep the thickest part of any piece no wider than about 1 inch.
Boil the salt, water, bay leaves, black peppercorns and chile for a minute or so, then turn off the heat and let this cool to room temperature. This gets the flavors melding. Once the brine has cooled to room temperature, pack the carrots and the thyme sprig into a clean Mason jar and pour the brine over them, making sure the jars all get some of the spices.
You will have leftover brine. Pour this into a plastic bag and tie it off. Push the bag into the jar — you want the carrots to be completely submerged in the brine. Alternately, fill a smaller jar that will just barely fit into the pickling jar with some water, screw on the lid and use that to prevent the carrots from contact with air. If the veggies hit air while fermenting, you get mold.
Put the jar into a cool, dark place for at least 3 to 4 days. What’s cool? Cooler than 75°F. Like many things that ferment, 55°F is about perfect. Don't go colder that 50°F. I ferment at about 72°F. You can leave the carrots in longer if you want — they will be saltier and tangier. I often ferment these carrots for a full 2 weeks.
Remove the bag or small jar from your pickling jar. To store your pickles, either do as I do and simply screw the cap on the pickles and put them in the fridge, or you can pour the brine into a clean pot and boil it. When it is cool, pour it back into the jar with the carrots and seal it up. If you want your pickles to be shelf-stable, you must boil the brine and then process it for 15 minutes or so in a hot water bath. Kept in the fridge, these pickles will last up to 6 months.
Note that prep time doesn't include pickling time.