Put a 9-by-5-by-3-inch metal loaf pan in the freezer to chill.
Put 3/4 cup of the cream and 1/2 cup of the peppermint puff candies in a medium saucepan over medium heat and cook, stirring frequently, until the candies are melted, about 3 minutes. Remove from the heat and stir in 1 cup of the chocolate chunks until melted. Cool completely.
Meanwhile, whip the remaining 2 cups cream in a large bowl with an electric mixer on medium-high speed until firm peaks form, about 2 minutes. With the mixer running, slowly pour in the condensed milk and peppermint extract and whip until combined.
Pour half of the condensed milk mixture into the chilled loaf pan, then top with half of the chocolate sauce and half of the remaining chocolate chunks and peppermint candy. Cover with the remaining condensed milk mixture and top with the remaining chocolate chunks and peppermint candy. Drizzle with the remaining chocolate sauce. Loosely cover with plastic wrap and freeze until solid and scoopable, about 3 hours and up to overnight. Serve in a cup with a candy cane if desired.
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