Preheat oven to 300 degrees F. In a large bowl, beat butter with an electric mixer on medium to high speed for 30 seconds. Add brown sugar, baking powder, and salt. Beat until combined, scraping side of bowl occasionally. Beat in egg and vanilla until combined. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour. Stir in toffee pieces.
Shape dough into 1-1/4-inch balls. Place 2 inches apart on an ungreased cookie sheet. Dip the bottom of a glass in granulated sugar and flatten each ball to about 1/4-inch thickness.
Bake about 20 minutes or until edges are firm but not brown. Transfer cookies to a wire rack; cool. Makes about 48 cookies.