The caterpillars of the grass emerald (Pseudoterpna pruinata) feed on gorse, broom and petty whin and so your best chance of seeing it in Dorset is on the extensive heathland areas. Given its food plant and likely habitat calling it the grass emerald seems a little strange!
This is a pretty, delicate moth that is a delightful shade of pale green when newly emerged, hence the name emerald, but the green fades as it ages and can become grey over time which might make identification harder unless you good a good look at it. It is considered a common day flying species but it tends to rest during the day and unless flushed, when it will then make a short flight to another plant and rest again, it may well be overlooked. Indeed, despite spending many hours on heathland I have only encountered it once in exactly the circumstances described above.
The adult is on the wing from late June through until August and the small green and pink caterpillar emerges in late July onwards. It is somewhat unusual in that the caterpillar hibernates during the winter.
Pseudoterpna pruinata: the grass emerald moth