Generously oil a sheet pan (preferably one with sides) measuring at least 11 by 17-inches with vegetable oil. In a medium-sized heavy saucepan, combine the water, sugar, cream of tartar and corn syrup and bring to a boil over medium heat. Using a candy thermometer to test it, boil the mixture until it reaches 340 to 350 degrees. The color should be deep golden brown.
Remove from the heat and whisk in the butter, then mix in the sesame seeds and baking soda.
Pour the mixture onto the oiled pan and spread it out a bit with the back of a wooden spoon, to about a 1/4-inch thickness. For thinner brittle, place a silpat on top and roll to desired thickness while it's still warm.
Let harden, uncovered, in a cool place, 30 to 45 minutes. (To wash the saucepan, soak it overnight.) Using your hands, wearing white gloves if desired to keep off any fingerprints, break the brittle into pieces. Store in an airtight container. To rewarm, place in a 350-degree oven.;
