It is extremely hard to control the rate of fruit production precisely in a healthy manner. Pruning (good pruning cuts) in general encourage fruiting and over-pruning the tree or bad cuts can cause the tree to go into shock and not produce any fruit.
I would suggest thinning as an alternative. Thinning is when you selectively remove a few fruits to encourage the rest in a bunch to grow bigger and sweeter. You needn't wait for the fruit to form either, as you can thin early by removing buds just before spring sets in (or when they start to bloom). This might actually be easier as you won't have the leaves coming in your way.
With this, you only remove the flowers (and hence the fruits) and leave the leaves intact, thereby providing you with the same shade that it does now.
Mike Perry has made a couple of suggestions that build on the above:
Prune once every 2 to 3 years immediately after flowering has finished:
Open up the centre of the tree, get some air movement in there. Doing so will help prevent diseases getting hold.
Reduce its height, this will make future tree/fruit maintenance and fruit picking a little bit easier.
When thinning, it's generally recommended to remove 50% of the fruit (very early in the fruit growing season, probably best to do around the same time as pruning or shortly afterward). Yes it's labour intensive and may seem like a lot to remove, but the benefits of doing so greatly outweigh not doing so.
An additional thought about pruning. It might make the job a little more manageable seeing as we're dealing with 11 fruit trees in total:
Year 1: Prune 3 or 4 tress.
Year 2: Prune a different 3 or 4 tress.
Year 3: Prune the remaining 3 or 4 tress.
Then repeat that cycle ie Each tree is on a 3 year pruning cycle. Unfortunately there is no way round the annual thinning of all the trees...
Have you considered contacting local food banks, churches, charitable organisations, nursing homes, plum wine makers, etc to see if they will come and take this fruity glut off your hands?
Also if you have the time and means, drying plums are a great way of storing them...