Heat the oil in a medium saucepan over medium-high heat until shimmering. Add the onions and cook, stirring occasionally, until they are softened, 5 to 7 minutes.
Add the garlic and cook, stirring constantly, until fragrant, about 1 minute longer.
Add the oregano and tomatoes to the saucepan. Season with salt and pepper. Mash the tomatoes with the back of a spoon, or a potato masher, until the mixture turns into a chunky sauce.
Stir in the chicken or vegetable stock. At this point, if you prefer the broth to be smooth, use an immersion blender to blend until smooth, or blend in batches using a standard blender. Bring the mixture to a boil, then stir in the orzo and bay leaf. Cook, stirring frequently to keep the orzo from clumping, until the orzo is tender, about 10 minutes.
Remove and discard the bay leaf, then serve the soup with crumbled feta.
