In a small bowl, whisk the olive oil, vinegar and mustard. Season the dressing with salt and set aside.
In a large pot of boiling salted water, cook the green beans until crisp-tender, about 3 minutes. Using tongs, transfer the beans to a strainer; return the water to a boil. Add the potatoes and cook at a low boil until just cooked through, 15 to 20 minutes. Drain and toss in a large bowl with 2 tablespoons dressing. Cut the beans into 1-inch pieces and add to the potatoes; set aside.
Meanwhile, preheat a grill pan to medium. Lightly brush the chicken with vegetable oil and season with salt. Grill the chicken, turning once, until browned and cooked through, 6 to 8 minutes on each side. Let cool and cut into bite-size pieces.
Add the chicken, tomatoes, olives, capers and remaining dressing to the potatoes and green beans; toss. Add the lettuce and toss. Top with the eggs and serve with the bread and butter.