Combine the cream and milk in a medium saucepan.
Bring to just a simmer over medium heat, stirring occasionally, then remove from heat.
Meanwhile, combine the eggs, egg yolks, and sugar in a medium sized mixing bowl.
Use a hand mixer (mid speed) to beat until the mixture is thick, smooth, and creamy (about 2 minutes).
Measure out about a cup of the warm milk mixture and, with the hand mixer on low speed, add the milk mixture in a steady stream to the sugar/egg mixture. (Adding it slowly tempers the cream. If you were to add the eggs straight to the hot milk, your eggs would probably cook -- egg drop soup style.)
You can add another cup of the milk mixture to the eggs in a slow drizzle, just to be safe.
Pour the entirety of the mixing bowl's contents back into the saucepan and stir to combine.
Cook, stirring constantly, over low heat until the mixture is thick enough to coat the back of a spoon -- a few minutes. If you cook this too hot or too long, you will end up with LUMPY CUSTARD (which I will name my punk band some day -- I called it). If you end up with LUMPY CUSTARD, see "A Note About Lumpy Custard" below.
Transfer the custard mix to a mixing bowl and let it cool on the counter for a few minutes.
Stir in the applesauce until combined. This does not have to be thoroughly blended, because some chunky pockets of applesauce in the ice cream will be pretty tasty.
Melt the caramels in a small saucepan with 1/4 - 1/2 cup of heavy cream, stirring constantly. The more cream you add during the melting process, the thinner the caramel sauce will be. I used about 1/3 cup heavy cream.
Once melted, stir the caramel sauce into the custard base.
Cover the mixing bowl with cling wrap and put it in the fridge until it is completely cooled (a couple of hours at least).
Remove from fridge and freeze according to your ice cream maker's directions. I use a Cuisinart counter-top ice cream maker. It took about 30 minutes to freeze this to a soft-serve consistency.
When the ice cream is ALMOST frozen to soft-serve consistency, add the piñons then let it go a bit longer.
For a more solid ice cream, you can move the ice cream to a freezable container and leave in the freezer for a couple of hours to harden up.