Place a rack in the upper third of the oven and preheat the oven to 375 degrees F. Coat a large baking pan with the oil. Mix together the black pepper, cayenne, and thyme in a small bowl.
Mix the egg whites in another bowl until foamy, then whisk in the Worcestershire sauce, hot sauce, and salt. Add the nuts and toss to coat. Add the pepper mixture and toss to coat.
Pour the walnuts into the prepared baking pan, spreading evenly. Bake for 4 minutes, stirring the nuts to break up any clumps. Bake until golden brown, watching closely to prevent burning. Baking time is about 10 minutes total. Remove from the oven and pour into a bowl. Cool to room temperature before serving.
