Preheat the oven to 350 degrees F. and line baking sheets with parchment.
Place the butter and brown sugar in the bowl of an electric stand mixer. Cream together on medium speed until smooth.
Add the honey, egg, and vanilla extract. Stir together on medium speed until incorporated.
Add the graham flour, cornstarch, baking soda, and salt, and mix on low speed until combined. Add the flour, about ⅓ cup at a time, stirring until the dough gathers itself into a ball and pulls away from the sides of the bowl.
Between 2 sheets of parchment, roll the dough out to a thickness of about 3/16 inch.
Use a 2½ inch square cookie cutter to cut shapes, and transfer to the prepared baking sheets.
Use the end of a bamboo skewer to make holes (this will keep air bubbles from forming under the graham crackers as they bake.)
Bake for 14 to 18 minutes, or until golden. Graham crackers will be somewhat soft when warm, but will become more crisp as they cool.
Notes
If you don't have a stand mixer, a hand mixer may be used. But the dough will be very stiff, and you may need to knead the last bit of flour in by hand.