Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, beat the softened butter on medium speed until creamy. Gradually add the granulated sugar and sifted powdered sugar and beat until well combined.
Add the oil, eggs, and almond extract to the butter-sugar mixture and mix until smooth and uniform.
In a separate bowl, whisk together the flour, baking soda, salt, and cream of tartar.
Gradually add the dry ingredients to the wet mixture, mixing until just combined. Stir in the sliced almonds and almond toffee bits by hand.
Use a medium cookie scoop (about 2 tablespoons) to portion dough. Roll each scoop briefly in additional granulated sugar to coat, and place on the prepared baking sheets about 2 inches apart.
Dip a fork in granulated sugar and gently flatten each cookie in a criss-cross pattern.
Bake for 10–12 minutes, until the edges are set and lightly golden. Let cookies cool on the pan for a few minutes, then transfer to a wire rack to cool completely.