Step 1In a large, heavy skillet, heat the olive oil over medium heat until the oil is hot, then add the onions and toss to coat. Raise the heat and cook, stirring frequently, until the onions begin to color.
Step 2Reduce the heat to low and sprinkle in the sugar. Continue to cook the onions until they turn a deeper golden color, about 15 minutes.
Step 3Stir in the balsamic vinegar and continue to cook slowly until the onions are soft, darkly caramelized and sweet, about 45 minutes, stirring occasionally.
Step 4Season to taste with salt and set aside in a warm place until the peas are ready.
Step 1In a small skillet, heat the olive oil over high heat until hot but not smoking. Add the capers and fry, turning carefully, until they puff up and begin to pop. Strain the capers immediately with a slotted spoon and drain on paper towels. Set aside in a warm place until the peas are ready.
Step 1In a wide, deep pot heated over medium heat, add 4 tablespoons olive oil. Stir in the onions and garlic and sweat until the onions are softened, 4 to 5 minutes.
Step 2Stir in the yellow split peas and toss to coat in the oil, then stir in the chicken broth, water and one-half teaspoon salt. Cook, stirring occasionally until the peas have absorbed all the liquid and are the consistency of chunky mashed potatoes. Season to taste with salt.
Step 3Puree the peas with an immersion blender and add the remaining olive oil and lemon juice, to taste, if using. Continue to puree until the mixture is silky and smooth. This makes about 5 cups puree.
Step 4Spread the pea puree onto a large platter and top with the caramelized onions and fried capers. Serve immediately.