Is there a breed of sheep that can exist on only browse, no pasture? That's a very good question. Nothing naive about that.
Just to confirm, what kinds of products are your looking for from your sheep? Wool, meat and
milk? Or just one or two of those?
All sheep enjoy browse to some extent. The older breeds (like icelandic, shetland, any of the Finn descendants) more than most. I haven't yet encountered a breed that can live on only browse. Don't despair. There could be one I haven't learned about yet. There are sheep that live almost exclusively on
seaweed, so why not just browse? If they exist, it will probably be one of the nomadic breeds or a Finn decent. I can't imagine it being any modern breed.
Have a look at Natural sheep Care by Pat Coleby - I don't remember offhand if she mentions a breed that needs less or no pasture. If anyone knows, it will be her. At the very least, the book gives an excellent understanding of what can be achieved through nutrition. Everything from increase fertility, wool production, reduced parasites... &c and so on are strongly affected by nutrition.
There are a few challenges I can see with not having pasture, especially if you want good wool production. The nutrition the sheep received, especially the trace minerals, has a huge influence on their health and the products they produce. Any nutritional stress, and you can loose that year's wool production because of a weakness in the fibres. Like how humans' hair get's brittle and falls out more after an illness or emotional stress. Sheep are more susceptible to that sort of thing. A lot of the nutrients they need is in browse, however, the sheep can't necessarily absorb it into their system. Having a lot of browse seems to help their health... but exclusively...? I don't know. My personal feelings are that if it were my sheep, I would hesitate to have browse only unless the breed had a history of it AND my browse was similar to their historical norm.
Some other things eating grass does for sheep is that it helps regulate their temperature. In the winter, for example, having the right kind of hay or grass fermenting in their rumen produces the necessary heat to help them through the cold months. PH balance in the rumen is also very important to their health. Without grass, I imagine they would need some other way to keep it balanced. Bloat is one issue. Others involve being more susceptible to parasites (see Coleby above for the theory on nutrition and parasite management).
It may be, you would need to subsidize their diet with hay if they have no grass. Or possibly, goats might be
the answer.
Those are my thoughts. Don't be discouraged yet. I don't know everything. It will be interesting to see what thoughts others have.