There was that thing at that place in South Yarra where they served avocado on toast with a smear of Vegemite. "Trust us," the menu said. We did, and it was good, and now it is practically Australia's national dish, or at least a serious regional plate.

"Trust us," says the waitperson brandishing another menu, this time in East Melbourne, and this time with toast, peanut butter, heirloom tomatoes and toasted peanuts.

The peanut butter-tomato toast is an echo of a similar dish on the menu at Barry in Northcote, one of Square and Compass co-owner Kael Sahely's other successful cafes. The peanut butter is not house-crushed activated organic nuts, either, although the presence of other superfoods on the menu suggests it might be. "It's Kraft crunchy," says Sahely. It's  the peanut butter of our suburban childhoods.

The Kraft is sweeter and stickier than house-made, and it gives this a bit of flavour glue – some mouth-texture, some sugary-nutty contrast to the heirloom cherry tomatoes. File this one under tasty comfort food.

East Melbourne is a funny part of town, full of quiet streets lined with heritage buildings, a bit like a Liberal-voting North Carlton, and not famous for cafe-going. Square and Compass, the symbol of Freemasonry, is a nod to local heritage. The Freemason's hospital is just up the road, and it turns out the weekday crowds are full of lanyard-wearing local health-care workers (I half expected to see Asher Keddie in scrubs, or at least Don whatsisname).

But this is as far from a hospital canteen as you could get. The fit-out of the heritage terrace house is in cool creams and blacks, with a grano floor and timber tables, and a busy open kitchen.

The chef from Barry, Jeremy Fraser, is a co-owner here, and the menu has a lot in common with the list at its northern cousin: a good mix of healthy, tasty and tempting.

Avocado toast comes with goat's curd, raw beetroot relish and kale chips, and bircher gets zhoozhed up with fresh fruit and the crunchy activated buckwheat known as buckinis.

A broccolini bowl with avocado, spinach, activated almonds and poached eggs runs neck-and-neck in the Healthy Eating Stakes with a California superfood salad, that's a big bowl of kale, heirloom tomatoes, charred corn, goji berries and wild rice tumbled with black beans, quinoa and jalapenos.

Poached eggs with broccolini and cumin-spiced goat's curd comes with what looks like saffron threads on top. They turn out to be "chilli hair": deseeded, finely sliced and dehydrated for less heat and more sweetness.

At the luxe end of the menu is a rock lobster roll, a lightly toasted brioche bun, served open-sandwich style on a sparse white plate with a scatter of purple petals and loaded with sweet, sea-tinged South Australian crustacean, tangy grated green pawpaw and a slather of bitey sriracha mayonnaise.

A pulled-pork jaffle releases tangy apple sauce at the first bite, channelling childhood apple jaffles (remember them?), and then turns roast-pork-and-apple-saucy, with a creamy hit of fontina cheese and a side of mustardy cauliflower piccalilli.

There's good coffee, too: Seven Seeds for espresso, small Collingwood roaster Promised Land for filter. Just what the, er, doctor (groan) ordered.

Do… Come on the way to the footy – the MCG is a short walk south
Don't… Miss the killer flourless chocolate brownie with raspberry and labna
Dish… Poached eggs with spiced goat's curd and chilli hair
Vibe Offspring takes a lunch break