Combine curry paste, white wine vinegar, ginger, cumin, and cardamom in processor; blend to paste. Transfer spice paste to large bowl; add cut-up chicken pieces and rub to coat well, then sprinkle generously with salt and pepper.
Heat 1 tablespoon olive oil in large nonstick skillet over medium-high heat. Add chicken pieces and any spice paste remaining in bowl to skillet; cook until well browned, about 3 minutes per side. Transfer chicken to bowl. Add onions to skillet; sauté until golden, adding water by tablespoonfuls if dry, about 5 minutes. Add diced tomatoes with juice; bring to simmer. Return chicken pieces to skillet. Spoon some of juices in skillet over chicken. Bring to simmer. Reduce heat to medium-low; cover and cook until chicken is cooked through, adding water by 1/4 cupfuls to thin juices, if desired, and turning chicken once, about 25 minutes. Transfer chicken pieces and juices to shallow bowl. Divide chicken equally among 4 plates; sprinkle each portion with chopped fresh cilantro and serve.