Mix 1 teaspoon garlic paste, lemon juice, and salt together in a bowl. Place chicken breasts in a baking pan, spread evenly with the garlic mixture, and refrigerate 2 hours to overnight.

Preheat the oven to 400 degrees F (200 degrees C).

Preheat an outdoor grill for high heat and lightly oil the grate.

Pour oil on marinated chicken in the baking pan. Add 1 cup water and cover baking pan with a lid or aluminum foil.

Bake chicken in the preheated oven until no longer pink in the center and juices run clear, about 20 minutes. Drain and reserve juices in a saucepan.

Transfer baked chicken to the grill. Cook until grill marks have formed on all sides, 3 to 5 minutes.

Add remaining 1 cup water, remaining 2 teaspoons garlic paste, vinegar, tomato paste, paprika, black pepper, and ginger to the saucepan with the reserved chicken juices. Bring to a boil and cook sauce until slightly thickened, 3 to 5 minutes. Add additional tomato paste to reduce spiciness, if desired.

Dip cooked chicken breasts into sauce and use remaining sauce as gravy for side dishes.