Heat oil in a large skillet over medium-high until shimmering. Season both sides of chicken livers with 2 teaspoons salt and pepper. Cook until lightly browned, about 3 minutes per side. Transfer to a plate. Reduce heat to medium-low; add onions, 1/4 cup butter, and 1/2 teaspoon salt to skillet. Cook, stirring occasionally, until onions are deep golden brown, about 20 minutes, stirring in thyme and garlic during last 2 minutes of cook time. Remove from heat. Immediately add brandy and sugar, stirring and scraping to loosen browned bits on bottom of skillet. Let cool 20 minutes. Place eggs, onion mixture, livers, and remaining 11/4 teaspoons salt in a food processor. Process until a coarse puree forms, about 20 seconds. With processor running, add 1 cup softened butter, 2 tablespoons at a time, processing until blended and mostly smooth, about 1 minute. Spoon into an airtight container; refrigerate until slightly firm but spreadable, about 2 hours. Transfer to a serving bowl; garnish with parsley. Serve with baguette, hazelnuts, and jam.