They have just as much flavor as the leaves.
I used to think that herb stems were scraps. The best you could do was to compost them, but more than likely, I thought, they should just end up in the trash. Herbs are great, you use their leaves, end of story. Turns out, I could not have been more wrong.
Herb stems, like garlic skins, pack a ton of flavor. You can absolutely save and use them, a practice that will cut down on your kitchen waste and add to the flavor in many dishes. So you save money and make better soup. Who can be mad at that?
First, a word on which herb stems we're talking about. Really woody stems, like rosemary, thyme, and oregano are going to be harder to work with. We're concerned with "tender stems" from herbs, meaning the springy, green stems from plants like parsley, basil, and cilantro.
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Parsley is a major component of French cuisine. The bright green leaves add a nice accent to dishes that might otherwise be colorless. Parsley stems are part of the classic bouquet garni, the bundle of herbs that adds flavor to soups, stocks, and sauces. If you hang onto your parsely stems and bundle them up with thyme, peppercorns, and bay leaves in a bit of cheesecloth, you have the flavor basis for lots of classic French dishes.
Basil stems are just as fragrant as the leaves themselves, and you can mash them up or add them to pesto without fear. Cilantro stems can be worked into guacamole or classic green sauce. You can also freeze the stems with your other kitchen scraps and add them to your next batch of homemade stock. If you want, you can add them to olive oil and let the oil infuse with the herbaceous notes in the stems. Most applications where the flavor of the herbs is welcome but the leaves themselves aren't necessary for looks take nicely to herb stems. So use them up and never accidentally throw away all that flavor ever again.