My TTR came to me on Continental ContiEscapes which worked nicely on the tar. I've seen several others since on this pattern so they're popular. Generally the fatter the tyre, the slower the handling, so I wouldn't go too fat. You only need to go wide to put down more horsepower!
Coincidentally, I have had a quick look for tyres for my brother's TTR.
First off I misunderstood and thought he wanted really road-orientated tyres. I have always liked the Avon Distanzias for mostly road use and these have now been replaced by Trailriders which seem to have the same pattern.
Front tyre is £55 https://www.oponeo.co.uk/moto-tyre-details/avon-av53-trailrider-80-90-21-48-s-front-tl-m-s#240072786
Rear tyre is £77 https://www.oponeo.co.uk/moto-tyre-details/avon-av54-trailrider-120-80-18-62-s-rear-tl-m-s#239021957
He then said he wanted more 50/50 road/trail use and so I looked at Continental TKC 80 Twinduros.
Front £56 and Rear £77 https://www.oponeo.co.uk/moto-tyre/continental-tkc-80-twinduro
I hadn't seen the Continental ContiEscapes and they look interesting Simon! I wonder what they would be like for easy laning?
There are also the Bridgestone Trailwings to consider.
If you go along the 50/50 route I can't recommend Mitas e07s enough, had good enough handling to chase down much bigger road bikes on hairpins etc!.... pretty good and retains a bit of off road ability
My ultimate plan is to have a set of rims & tyres for off road & another set for mainly road use, but as there is no room to fit any more stuff (treasure) in my garage I will have to wait a while for that.