Allow the puff pastry to thaw for 45 to an hour. You need to time this pretty precisely—too little and it won’t unfold; too much and it will stick to itself. Turn the sheets of puff pastry out onto a parchment-lined baking sheet.
Meanwhile, cook, bake, or microwave the potatoes until done but still firm. Plunge into cold water.
Preheat the oven to 400º F.
Combine the greens, parsley, scallion, and optional dill in a large skillet or stir-fry pan. Wilt the leaves down over medium heat. This will only take a minute or so. Drain the greens and squeeze out as much of the moisture as you can.
When the potatoes are cool enough to handle, peel them and slice very thinly.
Place one puff pastry sheet on a parchment-lined baking sheet and fold out. Scatter half of the potato slices over the surface, cover evenly with half of the steamed greens, then sprinkle with half of the dried tomatoes. Grind pepper over the entire surface.
Roll the pastry over the filling, jelly roll-style. Roll so that the seam side is on the parchment. Tuck each end in slightly so the filling doesn’t escape. Make 10 shallow slashes over the top of the strudel at even intervals (about ½ inch apart) so that you end up with 10 or 12 sections.
Repeat with the remaining sheet of puff pastry and the remaining ingredients.
Bake for 15 to 20 minutes, or until the pastry is puffed and golden. Cool until just warm or at room temperature.
Slice through the sections you marked off with a sharp knife then transfer to a platter to serve.