Combine yogurt, mayonnaise, mustard, lemon juice and water in a small bowl.
Bring 6 cups of lightly salted water to a boil in a medium saucepan. Add green beans and cook until just tender, 4 minutes. Remove the green beans with a slotted spoon and divide among 4 plates.
Return the water to a boil; place eggs, one by one, in the boiling water and set the timer: 5 minutes for a soft-boiled egg, 8 minutes for hard-boiled. When cool enough to handle, peel and slice the eggs in half.
Meanwhile, heat the oil in a large nonstick skillet over medium-high heat, add polenta rounds in a single layer and cook, turning once, until crispy and golden, about 4 minutes per side. Place 2 polenta rounds on each plate and keep warm. Add the reserved sauce to the pan and cook over medium-low heat, stirring constantly to avoid scorching, until heated through, about 3 minutes.
Divide the polenta rounds among the plates, top with egg halves and drizzle with the sauce. Serve immediately.
