Whisk olive oil, sesame oil, brown sugar, 2 tablespoons vinegar, gochujang, and black sesame seeds together in a bowl. Cover vinaigrette with plastic wrap and chill.
Mix chicken and spinach together in a large bowl. Divide salad among 4 serving plates.
Fill a large saucepan with 2 to 3 inches of water and bring to a boil. Reduce heat to bring water to a gentle simmer; add 1 tablespoon vinegar and salt. Crack each egg into an individual bowl or cup then gently slip into the water at 10- to 15-second intervals. Cook eggs until the whites are firm and the yolks have thickened but are not hard, about 3 minutes. Remove the eggs from the water with a slotted spoon, draining water, and place one poached egg on each salad.