Make the sherbet:
Combine the sugar and water in a saucepan and bring to a boil. Let cool slightly, then whisk in the buttermilk and vanilla. Refrigerate until cold, then freeze in an ice cream machine.
Using vegetable oil, generously oil a sheet pan (preferably one with sides), at least 11 by 17 inches. In a heavy saucepan, big enough to accommodate the additional ingredients, combine the water, sugar, cream of tartar, and corn syrup and bring to a boil over medium heat. Using a candy thermometer, boil the mixture until it reaches 350 degrees F. Remove from the heat and, working quickly, whisk in the spices. Stir in the butter, peanuts, and baking soda. Pour the mixture onto the oiled pan and spread it out a bit with a wooden spoon, to about 1/4 to 1/2-inch thickness. Don't spread it too thinly. Let harden, uncovered, in a cool place, 30 to 45 minutes. (To wash the saucepan, soak it overnight). Using your hands, break the brittle into pieces. Store in an airtight container. Serve with scoops of sherbet.
As fancy new sorbets have taken over in the freezer case, poor sherbet has been shoved aside. Orange and rainbow flavors are about all you can find these days, and they're usually packed with dyes and additives. Sherbet deserves a little respect! Unlike fruit-and-water sorbets, sherbet contains a little bit of milk, which gives it a nice smooth texture, but only adds a speck of fat. Who would think it possible to take a pile of sugar and a pile of peanuts and spin them together into an shiny, chunky sheet of amber candy? If you've never made candy at home, try it and see how easy it is. Candy making used to be a popular American recreation. Taffy pulls, fudge parties, and praline-making all turned into festive, social events, especially in winter, when the warmth of the stove was most welcome. If you want to be a real perfectionist, wear cotton gloves while breaking the candy to prevent fingerprints. Salted peanuts are a better choice than unsalted here.