Mash butter, garlic, marjoram, and paprika in a small bowl to blend. Season marjoram butter to taste with salt and pepper.
Melt 1 tablespoon marjoram butter in a medium heavy skillet over medium heat. Season chicken with salt and pepper. Add chicken to skillet, cover, and cook until no longer pink in center, about 5 minutes per side. Transfer chicken to a plate. Tent chicken with foil to keep warm.
Increase heat to high. Add tomatoes to skillet and cook, stirring occasionally, until they begin to char and burst, about 5 minutes. Add remaining marjoram butter to skillet. Crush tomatoes slightly to release juices; stir 1 minute. Season sauce to taste with salt and pepper. Spoon tomato sauce over chicken. Garnish with parsley.