Preheat oven to 350° and line two baking sheets with parchment paper.
In a bowl, whisk together flour, baking powder, salt, baking soda, and cinnamon.
In bowl of a mixer fitted with the paddle attachment, beat butter and sugars on high speed until light and fluffy, about 3 minutes. Scrape down sides of bowl and beater. On low speed, mix in egg and vanilla. Add flour mixture and mix until just combined, then stir in oats by hand.
Roll dough into 23 or 24 balls (2 level tbsp. each). Arrange 6 balls on each baking sheet, 2 in. apart (remaining cookie dough balls can be frozen). Bake cookies, rotating pans halfway through, until golden brown and crisp, about 14 minutes. Let cool on pans.
Line a small baking sheet with parchment paper. Set 6 cookies on sheet, bottom sides up, and center 1 scoop ice cream (about 1/3 cup) on top of each. Press down on ice cream with back of a spoon to flatten it slightly, then top with a second cookie. Freeze until hard, at least 2 hours.
Pour room-temperature Chocolate Shell into a bowl. Working with 1 ice cream sandwich at a time, use a slotted spoon to lower it into Chocolate Shell, flipping to enrobe sandwich in chocolate. Lift out sandwich, letting excess chocolate drip off, and return to parchment-lined baking sheet. Repeat with remaining sandwiches and chocolate.
Freeze sandwiches until hard, at least 1 hour or up to overnight. (If storing longer, wrap each in waxed paper and put in an airtight container.)
*If ice cream is airy, you may need as much as 4 cups, because it will collapse when measured.
Make ahead: Up to 2 months, wrapped in waxed paper and frozen airtight.