Prepare the carrots: Rinse and dry the carrots (they must be completely dry). Cut them in half, then place the dry carrots in a large bowl.
Add the mustard seeds and fennel seeds to a spice grinder and coarsely grind.
Add the coarsely ground spices, salt, cayenne, and turmeric to the bowl with the carrots.
Add mustard oil to a saucepan over medium-high heat and cook for 4-5 minutes, or until you see the oil begin to smoke (you’ll want to turn the stove off as soon as the oil starts to smoke).
Take the pot off the stove and wait 20 minutes for it to cool down. Pour it over the carrots and spices and mix well.
Wait another 10 minutes, then stir in the lemon juice.
Once the mixture has cooled down completely, pour it all into a dry glass jar, lay the lid on top, but don’t close it fully. Let it sit on the counter for 5 days (that’s how long it takes to deepen in flavor). Shake it once a day so the oil coats all of the carrots.
After 5 days, move it into the fridge for up to 3 months. It will continue to improve in flavor as it sits. Before eating, remove some of the carrots from the jar, place them in a bowl, and let it come to room temperature, then enjoy.