Answer: The best time to prune flowering quince (Chaenomeles speciosa) is late winter or early spring. With an overly large, mature plant you actually do have two pruning options: to cut back some or to cut back a lot. One is to simply thin and prune branch by branch for size and shape using the standard renewal pruning techniques.
The other option is to cut the entire shrub off about six inches from the ground. If you do this, be sure the plant receives ample water during the following growing season. This technique is quite drastic but an otherwise healthy quince will recover and come back better than ever, although you will sacrifice bloom and fruit for the first year following this treatment.
In the meantime, if there are just a few branches in particular which are totally "in the way", it would be safe to trim them off now and then do the main pruning next winter. Be sure to wear plenty of protection (including for your eyes) when you prune, those branches are spiny!
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