Line a large plate with paper towels. Line a baking sheet with a silicone baking mat or parchment paper.
Pour the neutral oil into a medium saucepan and secure a deep-fry thermometer on the side. Set over medium-high heat and warm the oil to 325° F, being careful to maintain this temperature as you fry.
Carefully add the árbol chilies and fry, ladling oil over them constantly, until crisp, darkened in color, and shiny, 20–30 seconds. Using a slotted spoon, transfer the chilies to one side of the prepared plate. Add the anchovies to the pan and fry, stirring occasionally, until light golden brown and crispy, about 30 seconds. Transfer with a slotted spoon to the other side of the plate. Set aside and let cool.
Pinch the stems off the chilies and then, over a medium bowl, crush the chilies with your hands into pieces 1/3-inch or smaller. Add the chili flakes, salt, and peppercorns and mix to combine.
In a large saucepan over medium heat, combine the peanuts, sugar, and water and cook, stirring occasionally with a wooden spoon. The water will evaporate and the sugar will seize and become powdery and grainy, 10–12 minutes. At that moment, turn the heat to medium-low and keep cooking, stirring every minute or so, until the sugar melts and caramelizes to a dark blonde and the peanuts are evenly coated, 8–10 minutes more. If the mixture smokes at any point, take it off the heat for a few seconds and lower the heat before continuing. Remove from the heat and stir in the chili mixture.
Transfer the caramelized peanuts to the prepared baking sheet, spread into a single layer, and let cool, then break up pieces that are stuck together. Transfer the fried anchovies and caramelized peanuts to separate airtight containers and store at room temperature for up to 2 weeks. Serve the peanuts with the fried anchovies on top.