Place oven rack in centre of oven; preheat to 350ºF (180ºC).
With pastry brush and a little shortening, lightly grease 2 rimless baking sheets; set aside.
Place butter in large bowl. With electric mixer, beat at high speed until fluffy and light coloured. Add brown sugar and granulated sugar; beat until even fluffier. Beat in peanut butter.
Break 1 of the eggs into small bowl and add to batter; beat until blended. Break second egg into small bowl; add to batter along with vanilla and beat until smooth.
In separate bowl, whisk together flour, baking soda, baking powder and salt; pour half over batter. With wooden spoon, mix until no streaks remain. Stir in remaining flour mixture until no streaks remain.
Scoop dough by rounded tablespoonfuls (15 mL); roll into balls and place on prepared baking sheets, leaving 2-inch (5 cm) space between cookies. With fork dipped in flour, press cookies to make crisscross pattern and flattened to 1/2-inch (1 cm) thickness.
Bake, 1 sheet at a time, for 10 to 12 minutes or until firm and golden around edges and underneath. Wearing oven mitts, remove baking sheet from oven; with lifter, lift cookies off baking sheet and set in single layer on rack to let cool completely.
Repeat with remaining baking sheet of dough. Let first sheet cool completely before adding more dough so dough doesn’t melt.