Combine lemon slices, sugar, and halved garlic in a medium bowl.
Sprinkle chicken with 3/8 teaspoon salt and pepper.
Heat a large skillet over medium-high heat.
Add 2 teaspoons oil; swirl to coat the skillet.
Add chicken to pan; cook 4 minutes on each side or until done.
Remove chicken from pan to plate and keep warm.
Add remaining 1 teaspoon oil to pan; swirl to coat the skillet.
Add lemon mixture to skillet; cook 1 minute or until lemon slices are lightly browned, turning occasionally.
Return lemon mixture to bowl and keep warm.
Lightly clean pan with paper towels.
Heat pan over medium heat and add 1 1/2 teaspoons butter to pan; swirl until butter melts.
Add shallots, 1/2 teaspoon grated garlic, oregano sprig, and thyme sprig; and cook 1 minute.
Add vermouth or white wine to pan, gently scraping pan to loosen browned bits.
Bring to a boil; cook 3 minutes or until liquid almost evaporates and is reduced well.
Add remaining 1/8 teaspoon salt, stock, and flour to pan, stirring with a whisk.
Bring to a boil; reduce heat, and simmer 3-4 minutes or until liquid is reduced to about 2/3 cup.
Remove pan from heat; discard oregano and thyme sprigs.
Stir in remaining 1 1/2 tablespoons butter and capers, stirring until butter melts.
Return chicken and any juices to pan; turn to coat with sauce.
Top chicken with lemon mixture. Sprinkle with parsley.