Preheat the oven to 425°F (218°C) with a rack in the center position.
Chop the potatoes, leaving the skin on, into ½-inch chunks. Pat them lightly with a paper towel or clean kitchen towel to remove any excess starch. Heat 2 tablespoons of oil in a large (10-inch) cast-iron skillet over medium-high heat. Sauté the potatoes, tossing every few minutes, until golden brown on all sides. Season with salt and pepper and set aside on a plate for later.
Heat the remaining tablespoon of oil to the skillet and add the green beans and mini sweet peppers. Sauté over medium-high heat for 2 to 3 minutes, stirring frequently, until lightly charred. Add the corn kernels and tomatoes and sauté for an additional 2 to 3 minutes, stirring often. Add the garlic, and sauté for an additional 30 seconds or until fragrant. Add the potatoes back into the pan. Season the hash to taste with salt and pepper. Using a spatula or spoon, distribute the hash evenly in the pan in one layer.
Place the skillet in the oven and bake for 15 to 20 minutes, or until golden brown.
Meanwhile, fill a small saucepan with several inches of water, salt heavily, and bring to a boil. Reduce the heat so that that the water is barely simmering (small bubbles are visible on the bottom of the pan).
Crack the eggs, one at a time, into a small sieve set over a bowl. Discard any loose egg whites that drain into the bowl–and carefully transfer each egg into individual ramekins or bowls with a tight lip. Tip the eggs gently, try to do them simultaneously or quickly one after another, into the barely simmering water. Using a slotted spoon, gently make sure they are not sticking to the bottom of the pan. Cook them, turning them once, for 3 to 4 minutes, or until the whites are cooked through and the yolk is soft to touch. Lift with a slotted spoon and drain lightly on a paper-towel lined plate.
Remove the hash from the oven, distribute it evenly among plates, top the hash with the poached eggs and drizzle with basil oil, if using. Top with grated cheese, as desired. Serve immediately.
Tips for Success
Since the hash is cooked all the way through on the stove, you can skip the oven step in this recipe, but I love that it helps brown all the vegetables evenly (it helps make them a little bit more crispy) and keep the hash warm and out of the way while you prepare the eggs!