Preheat your oven to 350°F (175°C). In a mixing bowl, cream together softened butter and both sugars until fluffy—about 3 minutes. Beat in the egg and vanilla extract. In a separate bowl, combine flour, baking soda, cinnamon, nutmeg, and salt. Gradually stir the dry mixture into the wet ingredients, then fold in the oats. Spread the dough on a baking sheet and bake for 11-13 minutes until lightly browned. Let the cookies cool before breaking them into chunks.

In a medium saucepan over medium heat, combine whole milk, granulated sugar, maple syrup, and cinnamon. Stir constantly until the mixture becomes foamy and sugar is dissolved—this should take about 5 minutes. Once heated through, remove from heat and stir in heavy cream, vanilla extract, and coarse kosher salt.

Transfer the ice cream mixture into a freezer-safe bag or bowl and create an ice bath using a larger bowl filled with ice and water. Place the sealed bag in the ice bath for about 30 minutes, stirring occasionally. After cooling, refrigerate the mixture for at least 1 hour.

Once chilled, pour the mixture into your ice cream maker. Churn according to the manufacturer's instructions, typically about 15-20 minutes. In the final 2 minutes of churning, add the cookie chunks.

After churning, transfer the Cinnamon Oatmeal Cookie Ice Cream to an airtight container. Smooth the top and press a piece of plastic wrap against the ice cream's surface to prevent ice crystals. Freeze for at least 1 hour or until firm.