As with most simple dishes, the trick to this parmesan toast with a poached egg and arugula is in the execution: Brown the cheese on the toasts, dress the greens just so, and cook the egg so its yolk is soft and melds with the surrounding flavors upon serving.
Fill a medium pot with 2 inches of water and bring to a boil. Position an oven rack about 6 inches away from the heating element and heat the broiler on high.
Set the bread on a baking sheet, brush with 2 Tbs. oil, and sprinkle with the cheese. Broil until the cheese melts and browns, 1 to 2 minutes. Drizzle toasts with 1 Tbs. olive oil.
Meanwhile, reduce the heat for the water to medium low and add a generous sprinkling of salt. Gently crack the eggs into the water and cook until the whites set and the yolks firm to a soft consistency, 2 to 3 minutes. Use a slotted spoon to transfer to another large plate.
Add the arugula to a medium bowl and toss with the vinegar and 1/4 tsp. salt. Add 2 Tbs. olive oil and toss well.
To serve, layer a couple of slices of tomato, the egg, arugula, and pepper on the toasts.