Well, this is one square beyond your requested range, and based on years ago, but Miracle of Science made the best veggie burger I've ever had (and I'm omnivorous).
Oh, looks like I forgot! Central Square, full name "Miracle of Science Bar and Grill".
It's towards the MIT end of Central, (I've always assumed that's the reason for the name), and there's a Flour bakery rather near it, for dessert!
Although now that I think about it, I've got a friend who swears by the specialty burger at Alden&Harlow (Harvard Square, downstairs by the Brattle Theater). It's a full restaurant, so should have some vegan-friendly fare.
Possible downsides is I remember it being a bit $$$, and I'm not sure of the hours it's open. But if someone's a burger friend I'd take them there. Good luck!
Seems so obvious/basic but Cambridge Common does a terrific burger (big patty style, which may or may not be your thing but is absolutely mine) and has tons of vegetarian options.
@rednikki i’m more qualified to discuss the vegetarian requirement than the burger, but if you’re willing to go a bit farther afield, consider Veggie Galaxy in Central. retro diner food, vegetarian across the board, and a longtime personal favorite (iirc the founding chef used to work at Veggie Planet in Passim?)
also i wholeheartedly add my voice to the chorus endorsing Cambridge Common
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