Butterfly chicken breasts and pound until flat and thin. Set aside.
Add 1 tbsp butter and 1 tbsp olive oil to a medium pan, and heat on low until melted; add onion and cook slowly, stirring often, until lightly browned.
While the onions are cooking, add 2 tbsp butter and 2 tbsp olive oil to a large pan, and turn to medium heat.
Combine flour, parmesan, and salt on a plate or shallow bowl. Dredge chicken breasts through flour mixture until well coated. Add 3 chicken halves to the pan; cook until browned, about 4-5 minutes on each side. Remove from pan, and cook the other 3 halves.
Meanwhile, once the onions are browned, add the balsamic vinegar and sugar; stir thoroughly to ensure that all onion slices are coated. Cook for another 5 minutes; remove from heat.
To serve: place one piece of chicken on a plate. Smear a portion of the blue cheese wedge onto the chicken. Top with onions and pear.