In a 2 quart pot, combine heavy cream, milk, sugar & salt over medium heat until sugar completely dissolves, about 5 minutes. Remove pot from heat.
While mixture is heating, separate egg yolks into a bowl.
Slowly add about a third of the hot cream mixture into egg yolks, whisking constantly.
Add the egg mixture back into the remaining cream.
Heat the custard mixture on medium-low until the mixture has thickened enough to coat the back of a spoon, about 170°F.
Stir in butter flavoring and extracts.
Pour ice cream base into a shallow container and let cool slightly before covering the top with plastic wrap. This will help reduce any film from developing while it cools.
Chill in a refrigerator for 4 hours or up to overnight, making sure the mixture reaches 40°F before churning.
While the ice cream base cools, spread the cubed cake pieces on a sheet pan and freeze until firm.
When the ice cream base reaches 40°F transfer the mixture to an ice cream maker and churn for 10 to 15 minutes.
While the ice cream churns, mix the food coloring into the frosting with a whisk or a fork. This will soften up the frosting, making it easier to fold into the ice cream later.
After about 10 minutes, add the frozen yellow cake pieces & sprinkles to the ice cream mixture. Continue churning for 2 to 5 minutes.
Transfer the ice cream to a freezer-safe dish and place dollops of frosting on top.
Gently fold the frosting into the ice cream, making sure it doesn't get fully incorporated.
Cover the top of the ice cream with the reserved rainbow sprinkles & freeze for 4 to 6 hours before serving.