Pulsing fresh ginger and scallions in a food processor breaks them down into a fine mince, which allows them to gently cook in the hot peanut oil, mellowing their flavor without burning.
Process ginger in a food processor until very finely chopped, about 30 seconds, stopping to scrape sides of bowl as needed. Add scallions and salt; pulse until scallions are finely chopped, about 10 pulses, stopping to scrape sides of bowl as needed. Transfer to a medium-size heatproof bowl.
Heat peanut oil in a small saucepan over high until hot but not smoking, about 4 minutes. Carefully pour hot oil over ginger mixture (it will bubble up). Stir to combine. Let cool to room temperature, about 1 hour.
Sauce can be kept covered in the refrigerator up to 1 week. Let stand at room temperature before serving.