Heat a large skillet over medium-high heat and, when hot, add 1 tablespoon oil. When the oil is hot, add the peppers and saute until the skins blister and they are somewhat softened, about 1 to 2 minutes. Season the peppers with 1 teaspoon of the tamari and transfer to a plate. Cover and set aside while you prepare the beef.

Cut the beef into 8 slices. Place each slice of beef between 2 pieces of parchment paper and, using a mallet, pound until thin, approximately 1/4-inch thick. Season each slice with salt and pepper on both sides and divide the julienned red bell pepper and enokitake mushrooms among each piece of beef.

Tightly roll the vegetables into each slice and tie with kitchen string in 3 places so that they do not unroll during cooking. Quickly rinse the skillet with hot water or wipe clean with a paper towel.

Add the remaining 1 tablespoon of oil to the skillet and, when hot, add half of the beef rolls and cook, turning occasionally, until browned on all sides. Transfer to a plate and set aside, covered, while you cook the remaining beef rolls. Add the sake, mirin, sugar, and 1 1/3 cups water. Bring to a boil, then return the beef rolls to the skillet. Don't overcook the beef, it should still be pink inside when cut. Transfer the beef rolls again to the platter, and set aside, covered. Increase the heat to high, and continue to cook the sauce until it thickens, about 2 minutes. Add the remaining tamari and return the beef rolls to the skillet to coat with the sauce. Remove the kitchen string from the rolls and cut each roll into 3 or 4 diagonal slices before serving, with extra sauce ladled over the top.