Use the peeler to round the edges, and adjust the ends of the carrot to make the carrot hot dog a little more equal in size. If you are not motivated to do this, then all you have to do is peel the carrot and trim the ends.
I always start at the ticker end of the carrot. Start peeling in a rounded motion to round the end of the carrot. After you have rounded the end, start peeling from the thick end toward the thin end, as you are moving the peeler down the carrot, getting closer to the thinner end, release the pressure. The goal is to remove carrot from the thicker side to try to get the two sides to be kind of equal.
How to brine carrot hot dogs
Combine all of the brine ingredients in a pot, and bring to a simmer. Reduce the heat to low.
Heat a pan over medium-high heat. Add 2 Tbsp olive oil to the pan. Place the carrots in the pan, and cook the carrots until a deep golden brown on all sides.
Place the carrots in the brine. Cook at a low simmer for 20-30 minutes.
Using a paring knife, gently push the tip into the thickest part of the carrot. If the knife slides into the carrot with just a hint of resistance, the carrot is finished cooking.
Place the carrots and the brine in a container. Allow to cool, and place in the refrigerator for a minimum of 24 hours.
How to reheat the carrot hot dogs
Stovetop
Heat a pan heavy pan over medium-high heat, and add a dash of olive oil to the pan. Gently start to sauté the carrots until hot, the carrots maybe become a little crisp, which is great. Once hot, serve.
Grilled
Lightly brush the carrot hot dogs with olive oil. Place the carrot hot dogs on the grill. Place them in a location that is hot but is not too close to a flame. If the flame is too close the oil will burn and char the carrot dog in an unpleasant way. Continue to turn the carrot dogs until hot. Once hot, serve.
Oven
Preheat the oven to 425 degrees.
Lightly brush the carrot hot dogs with olive oil. Serve when hot.