Line a rimmed baking sheet with parchment and lightly grease it with neutral oil. Clear space in the refrigerator.
Wash apples in hot water, dry them very well, and remove any stems. Insert wooden craft sticks into the stem end of each apple. Refrigerate the apples while you make the caramel, 15–20 minutes.
In a heavy 3–4 quart saucepan, combine the unsalted butter, light brown sugar, light corn syrup, and sweetened condensed milk. Set over medium heat and stir constantly with a heatproof spatula until the sugar dissolves and the mixture is homogenous, 3–5 minutes. Attach a candy thermometer to the pan.
Bring to a steady boil over medium to medium-low heat. Cook, stirring constantly and scraping the bottom and corners, until the caramel reaches 245–248°F (firm-ball stage), 12–18 minutes; it will thicken, bubble slowly, and coat the spatula.
Remove from heat and stir in the salt and vanilla extract. Let the caramel stand 3–5 minutes to cool slightly to a thicker dipping consistency (about 190–200°F).
Working one at a time, dip a chilled apple into the caramel, tilting the pan and rotating the apple to coat evenly. Lift, let excess drip back into the pan, then gently scrape the bottom of the apple against the rim. If using chopped peanuts, immediately roll the caramel-coated apple in them.
Set the dipped apples on the prepared baking sheet. If the caramel becomes too thick, warm the pan over low heat and stir until fluid again.
Let the apples stand until the caramel is set and no longer tacky, 20–30 minutes. Serve the same day, or refrigerate up to 2 days and let stand 10 minutes before eating for a softer bite.