Using an electric mixer, cream the butter and sugar together until light and fluffy--3-5 minutes. (Don't skimp on the time--this makes a big difference in texture.) Add the tangerine zest, vanilla, and salt and mix in. Then slowly add the flour until just combined. Dump the dough onto a large piece of plastic wrap. Press into a disk, then wrap well. Refrigerate for at least 30 minutes before rolling.
Preheat the oven to 350 degrees F and line your cookie sheets with parchment paper. Break the dough in half, leaving half in the fridge. Flour a work surface and roll out half the dough until 1/8- to 1/4-inch thick. Using a 2- to 3-inch cookie cutter, cut out the cookies very close together. Then gather the remaining dough, re-roll and cut again. Every time you roll out the dough it gets tougher, so try to cut as many from the first roll as possible. Repeat with the remaining dough.
Bake for 15 minutes and cool on the cookie sheets 2-3 minutes, before moving to a cooling rack. Meanwhile...
For the Candied Tangerine Peels:
Score the peel of 3 tangerines in three complete circles. Make sure not to cut all the way through the fruit, so you can save it for snacks. Pull the peel from the stem end of the tangerines down to the navel end, forming long 3/4- to 1-inch wide strips. Cut the strips into thin slivers.
Place the rinds in a microwave safe bowl. Cover with water and microwave until the water is boiling, 2-3 minutes. Drain, refill, and repeat.
Stir the 1 1/4 cups sugar and 3/4 cup of water in a small saucepan over medium-high heat until the sugar dissolves. Bring to a boil. Add the tangerine peels, reduce the heat to medium-low, and simmer until tender, about 15 minutes. Using tongs, transfer the peels to a sheet of parchment paper to dry slightly, about 30 minutes. Place the remaining 1/2 cup sugar in a bowl and toss in the rinds to coat.
Place the chocolate chips in a microwave safe bowl. Microwave in 30 second increments, stirring in between, until the chocolate has just barely melted, usually 60-90 seconds total. Pour the melted chocolate into a zipbag. Close the bag and snip off a tiny corner on one end. Use the bag to pipe chocolate drizzle over each cookie. Then before the chocolate dries, place a piece of candied tangerine on top. Allow the chocolate to dry completely before moving.