Combine the milk, sugar, brown sugar, and salt in a medium saucepan over medium heat. Stir until sugar is melted and mixture is smooth. When it starts steaming and little bubbles appear around the edges, remove from heat. Add the cinnamon sticks, cover, and let steep at room temperature for 1 hour.

In another saucepan, melt the butter over medium low heat. Continue to cook the butter, stirring frequently, until golden brown. Do not burn. Remove from heat and set aside.

Whisk the egg yolks together in a large bowl. Remove the cinnamon sticks from the milk mixture and return to medium heat. Once it starts steaming, remove from heat and immediately pour it in a small stream into the egg yolks, while whisking constantly.

Return egg and milk mixture to the saucepan and place over medium high heat. Cook mixture until it coats the back of a spoon, about 5 minutes, stirring constantly. Remove from heat and add the browned butter a little at a time while stirring.

Pour mixture through a mesh strainer into a large bowl. Stir in heavy cream. Chill thoroughly in the fridge.

Churn in an ice cream machine according to manufacturer's instructions. Scoop ice cream into a container and place in freezer to harden for a few hours before serving.