Place the cherry tomatoes and the vegetable broth in a large pan; cover. Bring to a boil, then reduce the heat and simmer until all the cherry tomatoes have burst, about 10 minutes.
Remove the pan from the heat and allow the tomato mixture to cool slightly, about 10 minutes.
Working in batches, puree the tomatoes in a food processor or blender until smooth. Note: Even if you can fit all of the tomatoes and liquid into your food processor or blender, you still want to work in batches because you are working with hot food! You need to be very careful and take precautions since the hot food can build up pressure and cause the lid to come off of your machine, splattering hot food all over your kitchen (or you)! Always follow the manufacturer's instructions for your specific food processor or blender, but here are some general safety guidelines:For a blender, don’t fill the container more than half full. For a food processor, don’t fill the bowl past the maximum fill/maximum liquid line. Put the lid on securely. If using a blender, remove the cap in the center (the steam needs an escape route). Cover the lid with a folded kitchen towel (one that is large enough to cover the lid and protect your hands). Blend/process on a low speed while firmly holding the lid in place with the towel.
In the same pan that you cooked the tomatoes in, saute the onions in the olive oil over medium heat, stirring occasionally, until the onions have softened, about 4 minutes.
Add the garlic and saute, stirring constantly, for another minute.
Add the pureed tomatoes, dried basil, dried oregano, and salt and pepper to the pan. Bring to a boil, then reduce the heat and simmer until the sauce has reached the desired thickness (it usually takes about 10 minutes to get it to the thickness I like). Taste and adjust seasonings, if needed.