Place eggs in large saucepan, add enough cold water to reach 2 inches above eggs, and bring to a boil. Remove from heat, cover pan, and let sit 12 minutes. Drain eggs and return them to saucepan. Gently shake saucepan to crack eggs all over.
Place 4 drops food coloring in large bowl and fill halfway with water, adding additional food coloring to reach a very dark color. Add eggs, increasing water amount so eggs are covered. Refrigerate at least 2 hours and up to 6.
Drain eggs, rinse under cool water, and peel. Halve eggs lengthwise. Transfer yolks to small bowl.
Add mayonnaise, chimichurri, lemon juice, and ⅛ teaspoon each salt and pepper; spoon into egg whites and sprinkle with paprika if desired.
