Separately cut kumquats and shallot crosswise into thin slices, discarding any kumquat seeds. Season chicken with salt and pepper. In a skillet heat butter over moderately high heat until foam subsides and saute chicken, skin sides down, until skin is golden and crisp, about 5 minutes. Turn chicken over and cook over moderate heat until just cooked through, about 5 minutes more. Transfer chicken to a plate and keep warm, covered.
Add shallot to fat remaining in skillet and cook, stirring frequently, 1 minute. Sprinkle sugar over shallot and cook, undisturbed, until sugar is melted and golden. Immediately stir in kumquats, water, vinegar, and red pepper flakes and simmer, stirring, until sugar is dissolved. Stir in parsley and salt to taste and, if sauce is too liquid, simmer until thickened to desired consistency, about 1 minute.
Transfer two thirds sauce to a small bowl and to remainder add spinach and salt to taste, turning with tongs until just wilted.
Divide spinach between 2 plates and top with chicken. Spoon sauce over chicken.