Heat 1/4 cup of the olive oil in a large deep skillet with a lid over medium heat.
Place the potatoes in the pan cut-side down.
Add the garlic and cook until the potatoes are slightly golden and crisp, 4 to 5 minutes.
Add the escarole, season with salt and pepper, then add about 1/3 cup water. Cover and steam until the escarole wilts, about 5 minutes.
Season the fish with salt and pepper; place on top of the greens in the pan and sprinkle with the oregano leaves. Cover and steam until the fish is just cooked through, about 5 more minutes.
Meanwhile, whisk the lemon juice in a bowl with the remaining 1/4 cup olive oil; season with 1 teaspoon salt and plenty of pepper.
Carefully lift the fish off the greens and transfer to rimmed plates or bowls. Distribute the greens, potatoes and pan juices around the fish.