Preheat your oven to 350°F (175°C).
Cream together softened butter, brown sugar, and granulated sugar until light and fluffy, about 2-3 minutes.
Add in the egg and vanilla extract; mix until fully combined. Gradually incorporate flour, baking soda, cinnamon, nutmeg, salt, and oats, mixing until just combined.
Scoop dough onto a parchment-lined baking sheet, leaving space between each scoop. Bake for 11-13 minutes until edges are golden brown.
Combine milk, sugar, maple syrup, and cinnamon in a saucepan over medium heat, stirring until foamy but not boiling, about 5-7 minutes.
Remove from heat, and mix in heavy cream, vanilla, and coarse kosher salt.
Cool the ice cream mixture in an ice water bath for 15 minutes, then refrigerate for at least 1 hour.
Churn the chilled mixture in an ice cream maker, adding cookie chunks in the final moments.
Transfer the mixture to an airtight container and freeze for at least 1 hour before serving.