Place the peanut butter, garlic, ginger, soy sauce, brown sugar, sesame oil, black vinegar, and chili oil into the bowl of a mini-food processor. Process until the mixture is well combined and forms a paste, 1 to 2 minutes. With the processor running, gradually add the chicken broth and process until the sauce is creamy and well combined, 1 to 2 minutes. Transfer to a large mixing bowl, cover, and set aside while you prepare the rest of the dish.
Place 4 quarts water into a large pot and bring to a boil over high heat. Add the noodles and cook until al dente, 1 to 1 1/2 minutes. Drain thoroughly in a colander. Add the noodles to the bowl with the sauce and toss to combine. Serve topped with the peanuts and scallions.