Warm the milk, cream, and sugar. Combine the whole milk, heavy whipping cream, and ½ cup of sugar in a medium saucepan. Then, heat the mixture over medium heat, stirring occasionally, until it reaches 175°F (about 5 minutes). Do not let it boil.
Prepare the egg yolks. While the milk mixture heats, whisk the egg yolks and the remaining ¼ cup of sugar in a medium bowl. Whisk vigorously until the mixture is pale yellow and creamy.
Temper the eggs. Slowly add the warm milk mixture to the egg yolks, ½ cup at a time, whisking constantly to avoid cooking the eggs. Once half of the warm milk has been added to the egg mixture, pour the entire mixture back into the saucepan.
Cook the custard. Return the saucepan to medium heat and cook, stirring constantly with a wooden spoon or silicone spatula, until the custard thickens slightly and coats the back of the spoon (or reaches 185°F). Be careful not to let it boil.
Strain and chill. Remove the saucepan from heat and stir in the vanilla extract and a pinch of salt. Then, pour the custard through a fine-mesh strainer or cheesecloth set over a large bowl to remove cooked egg bits. Let the mixture cool to room temperature, then cover with plastic wrap and refrigerate until thoroughly chilled—at least 6 hours or overnight.
Churn the custard. Once the custard is fully chilled, set up your ice cream maker according to the manufacturer’s instructions. Then, pour the custard into the ice cream maker and churn until it reaches the consistency of soft-serve ice cream.
Add the candy canes. Next, slowly add the crushed candy canes, ¼ cup at a time, during the last few minutes of churning.
Freeze until firm. Transfer the churned ice cream into a freezer-safe container. Cover tightly and freeze for 2–4 hours until the ice cream is firm.
Serve and enjoy. Scoop the ice cream into bowls or cones and garnish with additional crushed candy canes, whipped cream, or a drizzle of chocolate.