Make the barley: Bring a medium pot of water to a boil. Add the barley, and cook until tender, 35 to 40 minutes. Drain well, transfer to a medium bowl and toss with 1 tablespoon of the butter and 1/2 teaspoon salt. Serve warm or at room temperature. (The barley can be made and refrigerated up to 2 days ahead. Microwave just enough to take the chill off, about 2 minutes, stirring about halfway through.)
Make the tomato-pepper sauce: While the barley cooks, blend the tomatoes, peppers, mayonnaise, tarragon, garlic and 1/2 teaspoon salt in a blender until creamy and smooth. Refrigerate until ready to use. (The sauce can be made and refrigerated up to 2 days ahead.)
Make the bacon and breadcrumbs: Put the bacon in a large nonstick skillet, and set over medium-high heat. Cook, stirring occasionally, until golden and crispy, about 5 minutes. Remove the skillet from the heat, and transfer the bacon to a paper-towel-lined plate with a slotted spoon. Pour off all but 1 tablespoon of bacon drippings, and return the skillet to medium-high heat. Add the breadcrumbs, and stir constantly until golden and crunchy, about 2 minutes. Transfer the breadcrumbs to a small bowl.
Fry the eggs: Wipe out the skillet, and melt the remaining 1 tablespoon butter over medium-high heat. Once the butter begins to foam, crack all 4 eggs into the skillet, and sprinkle each with some salt. Cook the eggs until the whites have set and the yolks are still runny, about 4 minutes. Remove the skillet from the heat.
Build the bowls: Divide the barley evenly among 4 bowls. Top each with neat rows or piles of bacon, frisee, goat cheese, olives and sauce (leave room for the egg). Top each with an egg, some sauce and a sprinkling of toasted breadcrumbs.
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