For the soup, place all of the ingredients, except the oil into a saucepan over a medium heat. Bring to the boil and reduce the heat to simmer for ten minutes, or until the broccoli is soft. Blend with a hand blender until smooth.
Meanwhile, for the scones, place the butter and flour into a food processor. Blend until the mixture has the texture of fine breadcrumbs.
Add the egg and salt and freshly ground black pepper and blend to combine. Continue to blend, gradually adding the milk, to make a soft, but not sticky, dough.
Shape the dough into rounds with floured hands and place onto a baking sheet. Sprinkle over the cheddar and place in the oven to bake for ten minutes, or until risen and golden.
To serve, ladle the soup into a bowl and drizzle over the oil and serve with a cheese scone.