Bring a large saucepan of water to a boil over medium-high heat. Using a slotted spoon, carefully lower the eggs into the water one at a time. Cook for 6½ minutes, adjusting the heat to maintain a gentle boil. Transfer the eggs to a bowl of ice water and chill until just slightly warm, about 2 minutes.
Gently crack eggs all over and peel, starting from the tip end, which contains the air pocket.
Meanwhile, make the dressing: in a mini food processor, pulse the anchovies with the capers, lemon zest, lemon juice and vinegar until finely chopped. With the machine on, drizzle in the melted ghee and the olive oil and puree until smooth. Season the dressing with 1/4 teaspoon salt and pepper to taste.
Arrange the chicories on a platter and drizzle with the dressing. Arrange the eggs around the perimeter and garnish them with coarse sea salt.