Combine ricotta and 1 tablespoon olive oil in a small bowl. Add salt to taste. Set aside.
Turn dough out onto a floured surface and divide into 4 pieces. Roll each piece into a 7-inch-long oval and brush both sides with olive oil. Place flatbreads directly on grill grates and cover grill. Cook, checking every 2 minutes, until dough bubbles and stiffens and flatbreads have visible grill marks on the bottom, about 4 minutes. Flip flatbreads, cover grill, and cook 1 minute. Remove from grill.
Spread about 1/3 cup relish on each flatbread. Dollop each with spoonfuls of ricotta mixture. Return flatbreads to grill, cover, and cook, checking every 2 minutes, until cheese is softened and dough is fully cooked, 3 to 4 minutes. Remove from grill.
Sprinkle with pine nuts, parsley, and oregano leaves. Drizzle with olive oil and balsamic vinegar to taste and dust with flaky salt.