Add the flour with the salt to a large bowl, then add the butter. Coat the stick of butter with flour in the bowl, then using a bench scraper cut the butter lengthwise in half, then lengthwise in quarters, coating each newly cut side in flour. Dice the butter and cover each piece in flour, then with a pastry cutter press the mixture as you would mash potatoes.
Add the ice cold water little by little and mix quickly with your hands just enough to create a ball of dough. Work the dough as little as possible.
Wrap the ball in plastic wrap and refrigerate for at least one hour.
While the pastry dough is resting in the fridge, prepare the filling.
Rinse the red potatoes and cut them into cubes. Do the same with the butternut squash.
Pour the extra virgin olive oil into a large pan, then add the crushed garlic clove and the sage, cut into thin strips.
Heat the olive oil for a minute, then add the squash and the potatoes. Season with salt and pepper and cook over medium heat, turning them often, until they are soft and golden.
Remove from the heat, transfer to a bowl and allow to cool completely.
When the potatoes and the squash are cold, add the diced pecorino and the pitted black olives, cut into rounds. Stir and set aside.
Heat the oven to 220°C (425°F).
Take out the pastry dough from the refrigerator and, with the help of a rolling pin, roll it out on a floured surface in a 30x40 cm sheet (12x15 inches).
Transfer the dough onto a sheet of parchment paper, this will help you to roll the strudel and transfer it to the baking sheet.
Spoon the potato and squash filling into the centre of the rolled-out dough, leaving about 2 cm on each side, then wrap the dough onto itself, starting from the long side.
Keep the open side on the bottom and seal the ends.
With the help of the parchment paper, transfer the strudel onto a baking sheet.
Brush the surface with a beaten egg and sprinkle with the seeds.
Bake the strudel for about 10 minutes, then lower the temperature to 375°C and bake for further 25 minutes. The strudel will be ready when golden and crisp.
Let the strudel cool down for about 5-10 minutes before serving, it will be much easier to slice it. It can be prepared in advance, even the day before, and warmed briefly in the oven before serving it.