In a medium saucepan, combine the heavy cream, whole milk, brown sugar, cinnamon, ginger, nutmeg, and salt. Heat over medium-low heat, stirring gently until the sugar dissolves, around 5-7 minutes.

Whisk the egg yolks in a separate bowl until well combined and slightly lighter in color. Gradually add about ½ cup of the warm cream mixture to the yolks, whisking constantly to temper them, about 2-3 minutes.

Return the tempered yolk mixture back to the saucepan with the remaining cream. Cook over low heat, stirring constantly until it thickens, reaching about 170°F, about 7-10 minutes.

Remove the custard from heat and stir in the vanilla extract and finely grated carrots. Blend until smooth.

Transfer the mixture to a bowl, cover with plastic wrap, and refrigerate for at least 4 hours or overnight.

Once chilled, pour the mixture into an ice cream maker and churn according to the manufacturer's instructions, for about 20-25 minutes.

Transfer the Carrot Cake Ice Cream to a freezer-safe container, pressing parchment or plastic wrap directly on the surface to prevent ice crystals. Freeze for about 4 hours until firm.

Allow to soften slightly at room temperature before serving for easier scooping.