Make the ice cream: In a medium saucepan, combine heavy cream and whole milk. Bring just to a simmer over medium heat. Remove from heat.
Whisk egg yolks and 3/4 cup granulated sugar together in a large bowl until pale and thickened, about 2 minutes.
Temper the eggs: Slowly pour about 1 cup of hot cream into yolk mixture, whisking constantly. Pour egg mixture back into saucepan with remaining cream.
Cook custard over low heat, stirring constantly, until mixture thickens enough to coat the back of a spoon, 5-7 minutes. Do not let it boil.
Strain custard through a fine-mesh strainer into a clean bowl. Stir in 1 tablespoon vanilla. Cover and refrigerate until completely cold, at least 2 hours or overnight.
Churn in ice cream maker according to manufacturer's instructions until soft-serve consistency. Fold in crushed Oreo chunks.
Spread ice cream evenly into a parchment-lined 8x8 or 9x9 pan. Press plastic wrap directly onto surface. Freeze until firm, at least 2 hours.
Make the cookies: In a stand mixer, beat softened butter, brown sugar, and 1/2 cup granulated sugar until light and fluffy, about 3 minutes. Beat in egg and 1 teaspoon vanilla.
In a separate bowl, whisk together flour, cocoa powder, 1/2 teaspoon salt, and baking soda. Gradually add to wet mixture on low speed until combined.
Turn dough onto plastic wrap and shape into a 2-inch diameter log. Wrap tightly and refrigerate for at least 1 hour (up to 3 days).
Preheat oven to 350°F. Line baking sheets with parchment. Slice dough into 1/4-inch rounds. Place 2 inches apart on sheets. Sprinkle with flaky salt if desired.
Bake for 9-11 minutes – cookies should look puffed and slightly underdone in center. Let cool on baking sheet 5 minutes, then transfer to wire rack to cool completely.
Assemble: Remove frozen ice cream block from pan using parchment. Cut into 12 squares with a hot knife. Place a square on the flat side of one chocolate wafer, top with another wafer, flat side down. Press gently. Repeat.
Place assembled sandwiches on a parchment-lined baking sheet and freeze for at least 1 hour to firm up before serving. Let sit at room temperature 2-3 minutes before eating.