For cookies:

Spoon cookie mix into a wire-mesh strainer set over a large bowl. Sift the cookie mix through strainer, leaving the chocolate chips in strainer (reserve the chocolate chips for another use). Add flour, eggs, oil, and butter flavoring to cookie mix in bowl. Stir until dough comes together. Shape into a ball, then wrap in plastic wrap. Chill dough in refrigerator for at least 30 minutes.

Preheat the oven to 375 degrees F. Line cookie sheet(s) with parchment paper and set aside.

On a lightly floured surface, roll out chilled dough to 1/4-inch thickness. Using a 3-inch flower-shaped cookie cutter, cut 12 shapes from the dough. Using a 1-inch round cookie cutter, cut out each flower center. Re-roll the scrap dough and cut out another 12 flower shapes, but do not cut out centers. Transfer the flowers to the prepared cookie sheet(s), spacing cookies 2 inches apart.

Bake cookies 8 to 10 minutes or just until starting to brown. Cool on cookie sheet(s) for 5 minutes. Transfer cookies to a wire rack to cool completely.

For filling:

In a medium bowl, stir together frosting and powdered sugar until well mixed. Gently stir in 1/2 cup toffee bits.

Spoon a heaping tablespoon filling onto each of the whole flower cookies. Top each with a cut-out flower cookie. Sprinkle a scant teaspoon of the remaining toffee bits in the center of each cut-out cookie.