Heat 1 Tbsp. oil in a large skillet over medium heat. Add the polenta slices and cook until browned and crisp on the bottom, about 5 minutes. Flip and cook until browned on the other side, about 5 minutes more.
Meanwhile, bring 2 inches of water in a wide pot to a boil. Add vinegar and reduce heat to medium-low to maintain a gentle simmer. Crack each egg into a small bowl and slip them, one at a time, into the simmering water, taking care not to break the yolks. Poach for 4 minutes for soft-set, 5 minutes for medium-set, and 8 minutes for hard-set. Using a slotted spoon, transfer the eggs to a clean kitchen towel to drain.
To serve, arrange 1 cup arugula on each plate. Divide the polenta slices and avocado between the plates and top each with a poached egg. Drizzle with the remaining 2 tsp. oil and season with salt and pepper.