Remove figs from syrup; cut each fig into quarters. Reserve 1/4 cup of syrup.
Heat water and sugar in a sauce pan over medium-high heat until sugar is dissolved and water begins to simmer. Reduce heat to medium; add white vinegar, red onions, yellow peppers, figs, and syrup. Return to a simmer and cook uncovered for 10 minutes, stirring occasionally.
Heat olive oil in a large skillet over medium-high heat. Brown chicken breasts, one at a time, until browned and the meat is no longer pink inside, about 4 minutes per side. Remove from pan to a paper towel-lined plate.
Mix cornstarch and water together in a small bowl. Add to fig sauce, stirring constantly until sauce is thickened.