Cut carrots in whatever way you desire, some people like them in matchsticks while others like them diced roughly. Fill a jar with the cut up-carrots, and the whole piece of ginger, whole garlic cloves, and the whole chili pepper that has been de-seeded.
In a separate measuring cup, dissolve the salt in 3 tbsp of hot water. Once dissolved, top up with room temperature water to 2 cups (500mL). Pour the room-temperature salt-brine into the jar until all the vegetables are submerged in liquid. Screw the lid on the jar. You may need to put a weight or some parchment paper on top to ensure that all the vegetables are submerged underneath the brine. Anything sticking out is at risk of getting moldy.
Allow to ferment for 5-7 days at room temperature on the kitchen counter. Keep away from direct sunlight. Burp the jar each day by loosening the lid to let out the CO2 gases that are produced from lacto-fermentation. This is important as the jar could explode if you do not release the pressure from the jar.
After 5-7 days, drain the vegetables from the brine. The brine can be saved and used as a starter culture for future ferments, or can be used to marinate meat in.
Roughly mince the lacto-fermented garlic cloves and ginger. Cut the chili pepper into thin strips. Mix back with the carrots.
Add the spices and mustard oil to the carrot, ginger, chili, and garlic mixture. Mix until all the vegetables are well coated in the spices and oil. Adjust seasoning based on your tastes. I usually add in more oil to create an oil barrier to help the achaar keep for longer. If you will eat it all in a few weeks, feel free to put in less oil. Transfer the achar to a clean jar, and place in the fridge.