Melt the butter in a 10-inch skillet over medium heat. Add the apples and season lightly with salt. Cover and cook, stirring occasionally, until very tender and caramelized, 10 to 15 minutes. If there is any liquid remaining in the pan, remove the lid and allow it to evaporate.
Purée the apples in a blender or food processor until very smooth. Taste the purée; it should be a little tart, but if it’s too tart, add a pinch of sugar. If it tastes a little flat, add another pinch of salt or a couple drops of fresh lemon juice, or both. Refrigerate until completely chilled, about 20 minutes.
On a lightly floured work surface, roll out one sheet of the puff pastry into a 12x14-inch rectangle. Brush off any excess flour. Orient the dough so a 12-inch side is parallel to the edge of your work surface. Spread a thin, even layer of the apple purée over the bottom half of the dough. Fold the other half of the dough over the purée so that the dough measures 12x7 inches. Trim the edges. Transfer the dough to a baking sheet lined with parchment and refrigerate until firm, about 30 minutes. Meanwhile, roll out the other sheets of dough and repeat the process. (It may not be necessary to use all of the apple purée.)
Lightly brush one of the chilled dough rectangles with the egg. Sprinkle 1/2 tsp. of the fennel and then 1/2 oz. of the cheese in an even layer on top. Season lightly with salt. Gently run a rolling pin over the surface to press the seeds and cheese into the dough. Chill while you repeat with the remaining dough, fennel, and cheese.
Working with one rectangle at a time, slide the puff pastry onto a cutting board. Using a pizza cutter, cut the dough lengthwise into long, thin strips about 1/3 inch wide. Working with one strip at a time, hold both ends and twist slightly (if the dough is too firm to work with, give it a few minutes to soften). Transfer the twists to 3 unlined heavy-duty rimmed baking sheets, spacing them about 1/4 inch apart and pressing the ends of each strip onto the pan to help hold the shape while baking. Cover and refrigerate until firm, about 15 minutes.
Position racks in the top, center, and bottom thirds of the oven and heat the oven to 425°F.
Put the cheese straws in the oven and reduce the heat to 375°F. Bake until golden, 20 to 23 minutes, swapping the sheets’ positions after 10 minutes. Turn off the oven, open the door, and leave the straws in the oven until very crisp, about 1 hour. Using a long metal spatula, carefully release the straws from the baking sheets. Transfer to a wire rack to cool.
These are best baked and eaten on the same day, but they can be shaped in advance on the baking sheets and refrigerated for up to 1 day or frozen for up to 1 week. Bake them directly from the freezer.