Preheat oven to 200 degrees Fahrenheit. Heavily grease two rimmed baking sheets, a bench scraper, kitchen shears, and (in the absence of kitchen shears) a very sharp knife. Line your work area with parchment paper or a Silpat (that's what I used) for the candy canes to cool on. Put one of the prepared baking sheets in the oven to warm.
Grab a large, heavy-bottomed saucepan and add to it the sugar, corn syrup, water, cream of tartar, and salt. Stir, then clip on your candy thermometer to the side of the pan. Do not stir again. Use a pastry brush dipped in water to wash down any sugar crystals from the sides of the pan, as needed. Cook the syrup until it reaches 310 degrees Fahrenheit (hard crack stage).
Get your gloves on!
Pour the syrup on the room temperature baking sheet and drizzle the peppermint oil over it. Use the bench scraper to work the oil into the syrup by scraping the bottom of the candy and folding it over the top. Do this several times, trying to distribute the peppermint oil throughout the syrup, before adding the vanilla and repeating. Keep cutting and folding until it becomes pliable, then cut in half. Put half on the heated baking sheet in the oven to keep it warm. This is the half that will be dyed later.
To make the white portion of the candy canes, you don't need any dye; the color will develop as you pull the candy (in fact, that will be an indication that you've pulled it enough).
Generously oil up your hands--it's time to get your work-out. Scrape up the candy into a ball and stretch it out into rope with your hands. Fold it over on itself, twist it, pull again, repeat. Go as quickly as possible--again, you can't let it cool off too much or you won't be able to work the candy (it will be hot, so you'll want to work quickly anyway). As you continue on, the candy will change color and gain a sheen. Once it has turned white, return it to its baking sheet and place in the oven to keep warm.
Remove the other half of the batch and pour the green dye over it. You'll need a lot, but you can adjust it to how pure you want the color to be. I used 1/3 of a tiny bottle of regular ole food dye. Use the bench scraper to work the color into the candy. You don't need to pull this half of the candy. Return to the oven for 5 minutes to warm it up.
Once the candy is warm enough to work with, remove both halves from the oven. Roll each half into logs and cut them into 4 pieces. Take a white and a green piece, then return the rest to the oven to keep them warm. (They may melt back into a blob again; just cut off pieces as needed.)
Line up the green and white pieces next to each other, press together, then begin twisting them, working as thin as you'd like. Use the oiled kitchen shears or sharp knife to cut every 4 inches. Work a hook into each section, pinch the ends, and place on the parchment paper/Silpat to cool completely. Continue working with the rest of the twisted candy.
Repeat with the remaining sections in the oven, working with a pair of pieces at a time. If they start to get too cool to work with, return them to the oven for a few minutes to soften up.
Cool completely before consuming (about 15 minutes). Wrap each candy cane with cellophane or plastic wrap and store in an airtight container. Candy will last for a few months.