Preheat the oven to 350°F. Do not grease the baking sheets (use ungreased cookie sheets).
Put the ½ cup granulated sugar for coating in a shallow bowl and set aside.
In a large bowl, use an electric mixer to beat 1 cup granulated sugar, 1 cup powdered sugar, 1 cup (2 sticks) softened butter, and 1 cup vegetable or canola oil until well blended and creamy.
Add 1 teaspoon almond extract and 2 large eggs to the butter-sugar mixture and mix until combined, scraping the bowl once or twice.
In a separate bowl, whisk together 3½ cups all-purpose flour, 1 cup whole wheat flour, 1 teaspoon baking soda, 1 teaspoon salt, and 1 teaspoon cream of tartar until evenly combined.
Add the dry ingredients to the wet ingredients in batches and mix on low speed until just blended and a dough forms. Do not overmix.
Stir in 2 cups coarsely chopped almonds and the one 6-ounce package butter toffee baking bits until evenly distributed.
Scoop about 1 tablespoon of dough per cookie and roll into balls. Roll each ball in the reserved ½ cup granulated sugar to coat.
Place the coated dough balls 3 inches apart on the ungreased cookie sheets. Dip a fork into the coating sugar as needed and press each ball with the fork to flatten slightly, making a criss-cross pattern. Re-dip the fork in sugar if it starts to stick.
Bake in the preheated oven 12 to 18 minutes, until the cookies are golden brown and crisp at the edges.
Remove the cookie sheets from the oven, let the cookies rest on the sheets for about 2 minutes, then transfer the cookies to a wire rack to cool completely.