Start by making the brownies. A box mix is fine. Make sure not to over bake them. Remove brownies from the oven and cool completely (if you are in a hurry stick them in the freezer). Cut brownies into bite-size chunks and set aside.
Meanwhile, in a saucepan combine chocolate chips and 3 tablespoons of butter. Cook and stir over low heat until melted.
Stir in 1 and 1/4 cups half and half or cream, 3/4 cup sugar, and corn syrup. Bring to a gentle boil over medium heat. Boil over medium heat, stirring frequently, about 8 minutes (set a timer!) or until sauce is reduced to about 2 1/3 cups. Remove from heat. Let cool for a minute or two, until it stops bubbling.
Stir in 1/4 teaspoon peppermint extract. Cool to room temperature; mixture thickens as it cools. (Freezer or fridge also helps here).
For the whipped cream, in a chilled mixing bowl combine 2 cups cream, powdered sugar, and 1/4 or 1/2 teaspoon peppermint extract. Beat with the chilled beaters of an electric mixer on medium speed until soft peaks form.
In 16 small dessert cups, alternately layer brownie chunks, chocolate sauce, whipped cream, and crushed candy canes, ending with whipped cream and candy canes. Serve immediately.
If you want to do this all in one trifle bowl, place 1/3 of the brownie chunks in a 3-quart bowl or trifle dish. Drizzle with 1/4 of the chocolate sauce. Sprinkle with 1/4 of the candy canes. Top with 1/3 of the whipped cream. Repeat layers twice more. Drizzle with remaining chocolate sauce; sprinkle with remaining candy canes. Serve immediately.