Make the ice cream base. In a medium saucepan, mix together the cream, milk, 1/4 cup of the brown sugar, and salt. Heat the cream mixture over medium-high heat. When bubbles form around the edge of the pan, take it off the heat.
In a medium bowl, whisk the egg yolks and remaining brown sugar until smooth.
Slowly whisk the hot cream (about 1/4 cup at a time) into the yolk mixture, whisking constantly.
Pour the yolk and cream mixture back into the saucepan. Put back on the stove and cook over medium to medium-low heat, stirring constantly, until the mixture thickens enough to coat the back of a spoon.
Strain the mixture through a fine-mesh strainer into a bowl. Add the vanilla. Set the bottom of the bowl in a larger bowl filled with ice and water until cool. Cover with plastic wrap and refrigerate 2 hours or overnight.
While the ice cream base is cooling, start the cookie dough bits. Preheat the oven to 350°F. Spread the flour on a parchment-lined baking sheet. Bake until it reaches 160°F, 5 to 9 minutes, stirring once or twice. Cool completely. Meanwhile, cream together the butter, both sugars and the salt.
Add the vanilla extract and cream to the bowl. Mix in the flour and stir until combined. Fold in the chocolate chips. Cover and chill for at least 30 minutes.
Line a baking sheet with parchment and drop cookie dough bits (about 1/2 teaspoon each) onto the sheet. Freeze until ready to use.
Freeze the ice cream base according to manufacturer’s instructions for your ice cream machine. When the ice cream is the consistency of soft serve, transfer it to a freezer-safe container.
Stir in the frozen cookie dough bits and freeze the ice cream until firm, approximately 2 hours.