In a medium saucepan over low heat, heat the heavy cream, milk, chocolate, and 1/4 cup of the sugar. Heat until mixture begins to steam and is scalding, but isn’t boiling.
In a separate bowl, whisk together the remaining sugar, egg yolks, and cocoa powder. Using a ladle, whisk in 1 cup of the heated cream mixture. Then, whisk the entire thing back into the main saucepan.
Continue to stir the mixture gently for around 5 minutes, until the mixture thickens and fully coats the back of a spoon. Remove from heat, stir in vanilla extract and salt, then place saucepan in an ice bath to cool for 5 minutes. Transfer mixture to a quart-sized ziplock bag and place in the fridge to cool completely, about 4 hours.
To make the cookie dough, combine the butter and sugars in a medium bowl. Stir until combined, then add in the flour, oil, and chocolate chips.
Cover with plastic wrap and place dough in fridge to cool for 30 minutes. After 30 minutes, remove from fridge and roll into centimeter-sized balls. Place back in fridge to harden.
Once the ice cream has cooled for 4 hours, churn in ice cream maker according to manufacturer’s instructions. Remove from mixer, fold in cookie dough pieces by hand, and scoop the entire thing into a large freezer-safe container.
Cover with plastic wrap and freeze for at least 8 hours, preferably overnight. Serve!