Heat the oven to 400°F and arrange a rack in the middle.
Pierce each potato several times with a fork or sharp knife. Place the potatoes directly on the oven rack and bake until the skins are crisp and a knife easily pierces the potatoes, about 50 minutes. Transfer to a wire rack until cool enough to handle, about 10 minutes. Set the oven to the broil setting.
Slice each potato in half lengthwise. Using a spoon, scoop out the flesh, leaving about 1/4 inch intact; reserve the flesh for another use. Brush the insides of the potatoes with the melted butter and season with salt and pepper. Flip the potatoes over, brush the skin sides with butter, and season with salt and pepper. Evenly space the potato halves skin side up on a baking sheet and broil until the butter foams and the skins start to crisp, about 2 to 3 minutes (keep a close watch so they don’t burn). Flip the potato halves over and broil until the top edges just start to brown, about 2 to 3 minutes more.
Whisk the oil and lime juice together in a medium, nonreactive bowl and season with salt and pepper. Add the arugula and toss to evenly coat. Divide the greens among the skins and top each with a heaping tablespoon of the pâté. Serve immediately.