The sauce can be made up to one day ahead and refrigerated in an an airtight container. Before serving, reheat over low heat; thin with coconut milk, as needed.
Place peanuts in the bowl of a food processor, and process until the consistency of fine meal. Set aside.
In a small saucepan, combine 2 cups coconut milk and the curry paste. Cook over high heat, stirring constantly to dissolve paste, until milk begins to steam.
Stir ground peanuts and remaining 2 cups coconut milk into saucepan, and bring to a full boil. Reduce heat to low; add lemon juice, sugar, soy sauce, and fish sauce. Simmer, stirring occasionally, until sauce has thickened and oil has risen to the surface, about 20 minutes.
Remove from heat. Let cool, and stir to reincorporate oil. Sauce should be the consistency of thick cream; thin with coconut milk, as needed. Serve warm.