Preheat the oven to 350 degrees F. Lightly grease an 8 by 8 by 2-inch pan.
In a mixing bowl, beat the oil, honey, granulated sugar, and egg until well blended. Stir the cinnamon into the flour, then stir this into the egg mixture and mix well. Stir in the nuts. Spread the batter in the prepared pan and bake for 25 to 30 minutes.
Remove from the oven and let stand while you make the glaze. In a bowl, stir together the confectioners' sugar, mayonnaise, milk, and vanilla. Pour the glaze over the cake and spread it with a metal spatula. Let cool, then cut into bars.