Wash and dry the apples very well. (See notes below.)
Pull the stems out of the apples and insert one end of a popsicle stick, caramel apple stick, or twig into the stem end, about halfway into the apple, depending on size. You want to leave enough of the stick available for gripping while also ensuring it's firmly in place. Return the apples to the fridge until ready to dip.
Line a baking sheet with a silicone mat (preferred) or with parchment paper sprayed with nonstick cooking spray, and set aside. If spraying with cooking spray, do NOT place the apples on this surface before coating with caramel.
Spread chopped peanuts on a plate and set aside.
Place the water, sugar, and salt into an impeccably clean 3- to 4-quart pot with a heavy bottom. Bring to a boil over medium heat, stirring constantly with a wooden spoon or heat-safe spatula until the sugar dissolves completely.
When the mixture boils, stop stirring. Simmer undisturbed until sugar syrup takes on an amber color, about 10 minutes. (For a long time the mixture will be clear, and you'll see the bubbles gradually get bigger. Once it starts to take on some color, the rest happens fairly quickly.)
As soon as the syrup is a nice, medium-dark amber, carefully pour in the cream. The mixture will boil up furiously. Stir vigorously with a wooden spoon or heatproof spatula. Continue cooking, stirring from time to time, until the caramel reaches 250°F (121°C).
At this stage you can choose whether to leave the caramel in the pot or pour it into a 2-cup heatproof glass measuring cup. Let the caramel cool to about 212°F (100°C).
One at a time, hold each apple by the stick and swirl to coat in the caramel. Let excess caramel drip back into the cup or pot. The caramel will cool quickly, so work as fast as you reasonably can. Place each apple on the prepared baking sheet. See notes below about reheating if necessary.
Roll the base of each caramel-coated apple in chopped nuts. Set each apple back on the baking sheet to cool completely.