Start off by whisking the tomato purée into the stock or water, then pour this into a medium-sized saucepan along with the pearl barley.
Bring up to simmering point and simmer gently for about 40 minutes or until the barley is tender. Meanwhile, chop all the vegetables very finely (as the pieces are going to be left whole and not liquidised, they do need to start off quite small). Ten minutes before the barley is ready, melt the butter in another saucepan, then stir all the prepared vegetables into it together with the garlic so they get a good coating of fat, then season with salt and pepper, turn the heat down to low, put a lid on and leave the vegetables to sweat to draw out the juices for about 10 minutes.
When you take the lid off the fresh aroma is fantastic! Now add the barley and stock to the vegetables and continue to simmer the lot (covered) for a further 10 minutes.
Meanwhile, make the sesame toast. Pre-heat the grill and toast the slice of bread on one side only. Then butter it on the untoasted side (right up to the edges) and sprinkle the sesame seeds all over it to cover as evenly as possible, pressing them down so they will stay in position (if you're lucky). Return the bread to the grill. Toast the sesame-encrusted side until the sesame seeds are lightly toasted and the bread itself is golden-brown. Then cut the slice up into four fingers.
Stir the cream into the soup and serve it very hot, with the parsley sprinkled over and with the sesame toast.