Preheat the oven to 350 degrees F. Spray a 13 by 9-inch pan with vegetable oil cooking spray and dust with flour.
In a medium bowl, beat the eggs with a whisk. Stir in the brown sugar until smooth. Pour into the top of a double boiler. Cook the mixture over simmering water for 20 minutes, stirring constantly, until it reaches 180 degrees F. Let cool for 5 minutes.
Sift the 1 1/2 cups of flour, baking powder, and salt together. Add pecans to flour mixture. Stir into the cooked mixture. Add vanilla and stir until combined. (The batter will be thin.) Pour the batter into the prepared pan and place the pan in the oven. Bake for 25 minutes.
Allow to cool completely in the pan before cutting into pieces. Store in an airtight tin in the refrigerator.