In a large saucepan, stir together the milk, heavy cream, salt, and vanilla. Bring the mixture to a bare simmer over medium heat.

While the saucepan is on the stove, vigorously whisk the egg yolks and sugar in a separate bowl until thick, ribbon-like, and slightly paled.

Whisking all the while, slowly add a few ladles of the hot milk mixture to the eggs. Add the entire egg mixture back to the milk on the stove and cook over medium heat, stirring constantly until the mixture lightly coats the back of a wooden spoon or a thermometer registers 180°F.

Strain the mixture through a sieve, allow it to cool slightly, then cover it and transfer to chill in the fridge until cold (around 3-4 hours, or overnight).

Preheat the oven to 350°F and line a baking sheet with parchment paper. Spread the flour in a thin layer over the parchment and bake for 5 minutes. Allow to cool slightly before use, or the chocolate chips will melt.

In a bowl, mix together the melted butter, brown sugar, granulated sugar, vanilla, and salt. Add the heat-treated flour and chocolate chips, then add the milk until the dough holds together well. Roll most of the dough into bite-sized pieces, reserving a bit for cookie crumble topping, if desired.

Take the ice cream base from the fridge and pour into a prepared ice cream maker. Churn according to manufacterer's instructions, then 5 minutes before ready, add in the cookie dough and additional chocolate chips (alternatively, you could finish churning the ice cream, then fold in the chocolate chips and cookie dough before freezing).

The ice cream should have a soft-serve consistency. Pour the mixture into a loaf pan or ice cream container, top with any remaining cookie dough, then cover and immediately place into the freezer to set for several hours. Once firm, scoop and enjoy!