Heat 1 tablespoon olive oil in a small skillet over medium-low heat. Add onion; cook and stir until softened and browned, about 5 minutes. Remove from heat.
Heat remaining 1 tablespoon olive oil in a large skillet over medium heat. Add chicken; cook until no longer pink in the center and the juices run clear, about 6 minutes per side. Remove from heat and shred using 2 forks.
Mix 1/4 cup cream cheese, hot sauce, vinegar, garlic powder, salt, and pepper into the shredded chicken.
Spread 1 tablespoon cream cheese over 2 focaccia slices; lay Cheddar cheese on top. Add heaping spoonfuls of shredded chicken mixture. Drizzle blue cheese dressing on top. Cover with onion and jalapeno rings. Top with remaining focaccia slices.
Preheat a panini press according to manufacturer's instructions. Cook sandwiches until Cheddar cheese is melted, about 2 minutes.