Preheat the oven to 350 degrees F.
In a medium bowl, whisk together the flour, baking soda, cream of tartar, and salt.
Using an electric mixer, cream the butter and sugars. Scrape the sides of the mixer bowl and beat in the oil, egg, and vanilla until smooth. Beat in the flour mixture in 3 small additions, scraping the sides of the bowl after each one. Slowly beat in the rice cereal, coconut, and oats. Fold in the toffee bits and refrigerate the dough, covered, for 1 hour.
Drop rounded tablespoons of dough on 2 cookie sheets, leaving at least 1-inch of space around each drop. Bake until the cookies are golden around the edges, 12 to 14 minutes. Transfer the cookies to wire racks to cool.
Using a double boiler, melt the white chocolate. Dip the cookies into the white chocolate and sprinkle crushed peppermint to decorate. Allow to cool or serve warm. Store in an airtight container at room temperature.