Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a medium bowl, whisk together the rolled oats, all-purpose flour, baking soda, ground cinnamon, and salt.
In a large bowl, cream the softened unsalted butter, brown sugar, and granulated sugar together until light and fluffy, about 3 minutes.
Add the egg and 1 teaspoon vanilla extract to the butter and sugar mixture and beat until combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Drop tablespoon-sized balls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake the cookies for 8 to 10 minutes, or until the edges are golden brown and the centers are set.
Remove the cookies from the oven and allow them to cool completely on a wire rack.
In a large mixing bowl, whisk together the heavy cream, whole milk, granulated sugar, and 1 teaspoon vanilla extract until the sugar is dissolved.
Pour the mixture into an ice cream maker and churn according to the manufacturer’s instructions until it reaches a soft-serve consistency, about 20-25 minutes.
Roughly chop the cooled oatmeal cookies into small chunks.
During the last 5 minutes of churning, add the chopped oatmeal cookie pieces to the ice cream maker to evenly distribute them.
Transfer the ice cream to an airtight container and freeze for at least 4 hours or until firm.
Before serving, let the ice cream sit at room temperature for 5 minutes to soften slightly for easier scooping.