Bathsheba, it should be possible to get your home-made burgers to stay in one piece whilst you cook them, if you form them then knead them together with your knife. I know it sounds odd, trying to get something to "hang together" by chopping it, but it does work.
When turning mine, I wait as long as possible, then slide the fish slice in under the one edge and push the burger with a fork from the other side of the burger up onto the fish slice. It isn't infallible but usually it works.
Left over meat is minced and used for shepherd's pie with chopped onions, and mashed potatoes. Vegetables are used up in omelettes, or put into much the same mixture as a Yorkshire pudding and baked in the oven, or used in soup, or in stir fry. We have never come to harm through eating left-over rice, which we steam until it is very hot, on the assumption that all left-overs should be warmed to boiling point to eradicate germs and preferably boiled for a couple of minutes.
My grannie made meat or vegetable pies with left-overs chopped up and put in a greased pie-dish with a puff pastry lid.