Pour enough oil into a medium pot so that it reaches a depth of 3 inches. Heat the oil until it registers 350°F (175°C) on a deep-fry or candy thermometer.
While the oil is heating, whisk together the garlic, anchovies, serrano chile, red wine vinegar, honey, scallions, walnuts, and extra-virgin olive oil in a bowl large enough to contain all the Brussels sprouts. Set the bowl near the stovetop.
Pat the Brussels sprouts COMPLETELY dry. Working in batches, carefully lower some of the Brussels sprouts into the oil and deep-fry until the edges begin to curl and turn brown, about 3 minutes. Dump each batch directly into the bowl of dressing and gently toss to coat.
After the last batch has been deep-fried, carefully add the parsley and capers to the oil—stand back, as the capers will pop and sputter—and then give the contents of the pot a stir. Remove the parsley and capers when the parsley becomes a deeper, more saturated shade of green, 30 to 60 seconds.
Add the parsley and capers to the bowl, season with salt and pepper to taste, and gently toss. Serve within minutes, if not seconds.