Place ginger, garlic, lemon juice, fish sauce and 1/4 teaspoon salt into a blender; add 2 tablespoons water and blend until smooth.
Place chicken pieces into a non-metallic bowl and pour ginger mixture over chicken.
Rub ginger mixture into the meat, cover bowl with plastic wrap, and refrigerate for at least 30 minutes (up to 24 hours).
In same blender pitcher (without rinsing), place tomato, cilantro leaves and stems, tomato paste, 3/4 tsp salt, and 2 tablespoons water. Blend until smooth.
Heat oil in a large, heavy nonstick pot with a lid over high heat. When oil is very hot, place chicken pieces with ginger marinade into pot; fry, turning chicken occasionally, until light brown on all sides, about 10 minutes. Pour cilantro mixture from blender over chicken. Cook, stirring often, until sauce thickens and oil separates, about 10 minutes.
Place coconut milk into pot, cooking and stirring until it is blended into sauce, 4 to 5 minutes.
Reduce heat to low, and cook until chicken is tender and no longer pink inside, 10 to 15 minutes.