Euonymus hedge

Although we clip our mature hedge,normally in September it has become far too thick and tall. It's probably 6' wide by 10' tall. I am happy to have the top cut drastically but would like to reduce the thickness.  I can't cut it in the spring as the birds nest there, I am not worried about losing the flowers.  Can I saw off some of the branches from the base and thin out others. My only worry is the plant dying, that would be disastrous. I realise it will look strange until it starts growing again.  Any help or advice appreciated.