Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a medium bowl, whisk together the rolled oats, flour, baking soda, cinnamon, and salt.
In a large bowl, beat the softened butter, light brown sugar, and granulated sugar together until creamy and smooth.
Add the egg and 1 teaspoon vanilla extract to the butter mixture and beat until combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Drop tablespoon-sized scoops of cookie dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 8 to 10 minutes, or until the edges are golden brown but the centers are still soft.
Remove from the oven and let the cookies cool completely on a wire rack.
In a medium saucepan, combine the heavy cream, whole milk, and 3/4 cup granulated sugar. Heat over medium heat until the mixture is hot and sugar is dissolved, but do not boil.
In a separate bowl, whisk the egg yolks until smooth.
Slowly pour about 1 cup of the hot cream mixture into the egg yolks, whisking constantly to temper the yolks.
Pour the tempered egg yolks back into the saucepan with the remaining cream mixture.
Cook over low heat, stirring constantly with a wooden spoon or heatproof spatula, until the mixture thickens and coats the back of the spoon (about 170°F or 77°C). Do not let it boil.
Remove from heat and stir in 1 teaspoon vanilla extract.
Pour the custard through a fine-mesh sieve into a clean bowl to remove any cooked egg bits.
Cover the custard with plastic wrap, pressing it directly onto the surface to prevent a skin from forming. Chill in the refrigerator for at least 4 hours or overnight.
Once chilled, churn the custard in an ice cream maker according to the manufacturer’s instructions.
Roughly chop the cooled oatmeal cookies into small chunks.
During the last 5 minutes of churning, add the oatmeal cookie chunks to the ice cream to mix evenly.
Transfer the ice cream to an airtight container and freeze for at least 2 hours to firm up before serving.