Preheat the oven to 200C/400F/Gas 6.
For the soup, heat the butter in a saucepan, add the onion and garlic and fry gently for five minutes.
Add the spinach and stock and simmer for 4-5 minutes.
Season, to taste, with salt and freshly ground black pepper, then use a hand blender to blend the soup until smooth.
For the garlic and sesame straws, roll out the piece of pastry on a lightly floured surface to make a 20cm x 15cm/8in x 6in rectangle.
Mix the beaten egg and garlic together in a small bowl, then brush the pastry with the egg mixture.
Sprinkle the pastry with the sesame seeds, then cut into ten long slices.
Place the straws onto a greased baking sheet and bake in the oven for 5-8 minutes, or until golden-brown.
To serve, pour the soup into warmed soup bowls and serve the straws alongside.