If you prefer not to use wine when cooking, substitute 3 Tbsp. lower-sodium chicken broth and 1 Tbsp. lemon juice.
Heat a large skillet over medium-high. Pound chicken breasts to an even 3/4-inch thickness. Sprinkle chicken evenly with 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper. Add oil to skillet; swirl to coat. Add chicken; cook until browned on both sides, about 4 minutes per side. Remove chicken from skillet.
Add wine to skillet; cook over medium-high, scraping skillet to loosen browned bits, until liquid is almost evaporated, 1 to 2 minutes. Add broth, beans, haricots verts, tomatoes, and remaining 1/4 teaspoon each salt and pepper; let mixture come to a boil.
Nestle chicken into mixture in skillet. Reduce heat to medium; cook, covered, until chicken is cooked through and haricots verts are crisp-tender, 9 to 10 minutes. Remove from heat; stir in butter, capers, and lemon juice.