Trash Heap
I keep a baggie in my freezer for all of my vegetable scraps; onion skins, parsley stems, dried out garlic, awkward ends-of-kale, yes, carrot tops. I keep a separate baggie of anything mushroom related, mostly stems. When I have a day off, and a full baggie, I put the refuse in a stock pot, cover it with water, and let it boil away until the water is full flavored, or I am very bored.
My mother once gave me one of those gigantic pasta pots with a colander insert which always seems like overkill for making pasta, but is perfect for making stock. When it's done I hoist the colander, let the veggies drain a while and that's it: perfect vegetable broth. If you don't have this kind of pot you can just pour your stock through a colander and into a separate pot or storage container. Someone more patient than me would strain this through a cheese cloth, but I don't mind the bits of vegetable refuse that make their way back into to liquid. This can be frozen for later use, but remember to leave a good amount of space in your containers as liquids expand in volume as they cool.
This week I made a roast chicken stuffed with Meyer lemons and thyme, and once it was carved, I threw it in a pot of water and let it simmer overnight. I had the next day off from work and slept in, so this was quite a long simmer. When I woke up I added my freezer scraps and the bowl of veggie ends you see above and let it all go another two hours. After straining, I added some cloves and a star anise and let it all reduce a bit. For dinner, I ladeled this stock over the left over torn chicken breast, and topped with bean sprouts and a mess of fresh herbs and a splash of fish sauce, for a very inauthentic (but very true to the leftovers-for-infinity approach to food-making that is central to Vietnamese cuisine) Pho.
A note on scraps: pretty much all of your kitchen scraps can get a second life as the base for stock. You might need to take consideration of strong flavors that might not blend well with others. For example, I save my fennel scraps for a special strong bone broth I make in the winter when I am sick as it adds a strange mentholated taste to the final stock, which is not unwelcome. Mushrooms get their own baggie in my case because I truly love mushrooms and make a very potent mushroom garlic stock that I use to make shitake soup and is amazing in risotto and braised bok choy as well. You can add yours to your veggie stock if you like.