Bring a large pot of salted water to a boil. Add beans, and cook until crisp tender, about 3 minutes. Drain, and immediately run under cold water to stop cooking. Drain well.
Heat grill to medium-high, and oil grates. Brush bread on both sides with oil; grill, turning once, until beginning to crisp, 2 to 3 minutes. Remove from grill, and let cool slightly. Rub 1 side of each piece of toast with garlic.
Meanwhile, slice green beans in half on the bias, and transfer to a large bowl. Toss green beans with half the vinaigrette and the chopped dill. Season with salt and pepper.
On a serving platter, break up burrata with a knife or a spoon. Flatten the cheese, and spread it out. Season with salt and pepper.
Pile green beans on top of burrata, drizzle with remaining dressing (or serve it on the side), and top with walnuts, a sprinkle of gochugaru if desired, and dill sprigs. Serve with grilled toast.
Recipe by Laura Rege.