1. Heat a large skillet over medium-high heat. Add oil to pan; swirl to coat. Sprinkle chicken evenly with 1/2 teaspoon salt and pepper. Add chicken to pan; sauté 6 minutes on each side or until done. Remove chicken; let stand 8 minutes. Slice and keep warm.
2. Reduce heat to medium. Add bacon to pan; cook 6 minutes or until crisp, turning once. Remove bacon from pan, reserving drippings; drain. Crumble bacon. Add shallots to drippings in pan, and cook for 30 seconds, stirring constantly. Stir in vinegar, scraping pan to loosen browned bits. Stir in broth, syrup, and mustard; cook for 1 minute, stirring frequently. Remove from heat. Arrange 1 1/2 cups arugula on each of 4 plates; sprinkle evenly with the remaining 1/8 teaspoon salt. Top each serving with 1 chicken breast half. Drizzle 2 tablespoons dressing over each serving; top with crumbled bacon.