In a saucepan, combine the milk, heavy cream, and sugar. Heat over medium heat, stirring until the sugar is completely dissolved. Do not boil.
Add the honey and gingerbread spice mix. Cook over low heat, stirring constantly, for 5 minutes.
Remove the saucepan from the heat and stir in the vanilla extract.
Let the mixture cool completely at room temperature, then refrigerate for at least 2 hours to chill thoroughly.
Pour the chilled ice cream base into an ice cream maker and churn according to the manufacturer's instructions. If you don't have an ice cream maker, pour the mixture into a freezer-safe container and stir with a fork every hour to prevent ice crystals from forming.
Once the ice cream begins to firm up, fold in the broken gingerbread cookie pieces to add a crunchy texture.
Transfer the ice cream to the freezer for at least 4 hours to harden completely.
Before serving, let the ice cream stand at room temperature for a few minutes to soften slightly and make it easier to scoop. Garnish with extra gingerbread cookie crumbs or a drizzle of honey, if desired.
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