Preheat the broiler.
In a large, oven safe skillet, heat the olive oil over medium heat. Add the potatoes, onion, and salt. Bring the oil to a gentle bubble, then reduce the heat to medium-low and cook, stirring often, until the potatoes are fork tender, about 20-25 minutes.
Using a slotted spoon, transfer the potatoes and onion to a bowl. Add the eggs, basil, dill, chives, and a large pinch of pepper, gently stir to combine, being careful not to break the potatoes.
Strain the oil out of the skillet and wipe the skillet clean. Heat 2 tablespoons of the reserve oil over medium high heat. Add the potato and egg mix and cook, stirring gently until the eggs just begin to set, about 2-3 minutes. Spread the mix into an even layer and reduce the heat to medium-low. Cook, shaking the pan occasionally to prevent sticking until the eggs are set and just cooked through, about 15-20 minutes. Transfer to the oven and broil 2-3 minutes.
Remove from the oven, run a spatula around the edge of the skillet to help release the tortilla, then carefully invert onto a serving plate. Let sit 10 minutes. Top with prosciutto, burrata, arugula, and fresh herbs. Slice and serve warm or at room temperature.
