Whisk olive oil, red wine vinegar, lemon juice, sugar, grated onion, seasoned salt, and basil together in a bowl to make dressing. Transfer to the refrigerator.
Combine water and chicken bouillon cubes in a saucepan. Heat over medium heat, stirring until bouillon cubes are dissolved, about 5 minutes. Add chicken breasts and onion; simmer until chicken is no longer pink in the center, about 45 minutes.
Transfer chicken to a cutting board using a slotted spoon. Reserve broth in the saucepan. Cool chicken until easily handled, 5 to 10 minutes. Chop coarsely.
Pour several cups of water into the reserved broth in the saucepan; bring to a boil. Stir in vermicelli and cook until tender, about 6 minutes. Drain.
Mix dressing, chicken, and vermicelli together in a bowl. Add artichoke hearts and toss to combine. Refrigerate until flavors combine, 2 to 6 hours. Mix in tomatoes before serving.