Preheat the oven to 450 degrees F. Line a rimmed baking sheet with parchment.
Toss the carrots with the olive oil and ¼ tsp. of the salt to coat. Arrange them in a single layer on the baking sheet, cover the baking sheet tightly with aluminum foil, and roast until tender, about 20 minutes.
Remove the foil, sprinkle the paprika and the remaining ¼ tsp. salt over the carrots, and toss to coat again. Transfer the carrots to a plate. Keep the oven on because you'll be using it again. Discard the old parchment and place a new sheet of parchment on the baking sheet.
Place the pastry sheet on a lightly floured surface and roll it out into a rectangle approximately 11 by 15 inches. With a paring knife, cut the pastry into long thin strips that are about 3 inches long and 1 inch wide, depending on the size of your carrots. A little bit of carrot should be able to stick out of each side once wrapped.
Wrap each carrot in a piece of puff pastry, seal the edge with a little water, and place it, seam side down, on the parchment. Brush the tops of the snuggies with water and sprinkle with sesame seeds, gently pressing the seeds into the pastry.
Pop them back into the oven until golden and puffy, 10 to 15 minutes. Serve with mustard.