Pizza shops are running a new game in Melbourne, adding regional American styles, ultra-deep bases and even toppings inspired by pasta sauce to their menus.

Connie's, the pizza joint tucked inside Heartbreaker Bar, hasn't been doing much late-night trade this year so its team have side-stepped into the square pizza game. Seven mammoth pies debuted this week, clocking in at about seven centimetres deep and 30 centimetres on all sides, with flavours including pepperoni, artichoke with alfredo sauce, and the Hawaiian-esque Don't Tell Nonna. The booze cabinet is also unlocked, so you can order a nip of rye whisky or a six-pack of Millers Lite and remember nights gone by at the divey CBD favourite.

Connie's is calling its new range "grandma pies" after the Long Island-style that sees pizza baked in rectangular pans, proofed for a short time and then layered with cheese followed by sauce. In fact, Connie's pizza is closer to the Detroit-style, known as much for its square shape as for its depth, which is similar to focaccia. Either way, it's an OTT contrast to the minimal Neapolitan pies Melbourne is accustomed to.

Carlton North's Capitano has served Detroit pizza for much of its three-year life and has expanded the range while we've been in lockdown. Right now, you can order a classic grandma (three different cheeses on red sauce) or pork sausage with calabrian chilli, plus three others. Each has deliciously brittle crispy bits lacing the edges.

Chef Casey Wall says a potato, four cheese, spring onion and pancetta square pizza that began as a special has "gone crazy".

"People were ordering twice on the weekend, then messaging us on Monday to ask if it would be back."

Wall originally wanted to serve only square pies at Capitano, but a mishap with the specialty pans he purchased from the USA meant the round shape had to come into play. Now the restaurant's customers order a fairly even split of both styles.

Can't decide whether you want pizza or pasta? A handful of pizzerias are solving the problem with toppings inspired by classic pasta sauces, such as cacio e pepe. That's been spotted at Freddy's in Windsor, where the team employ three cheeses – the traditional pecorino plus parmesan and fior di latte – to recreate the silky mouthfeel of the classic Roman sauce.

Carbonara pizza is also on the menu at Freddy's, as well as Richmond spots Baby and Ganzo and at Capitano. Wall says the idea came from wanting to put egg on a pizza, "but not like an Australiana pizza". In Wall's take, egg yolks are added to a base of pecorino cream, scamorza and guanciale after the pizza is pulled from the oven. An assertive amount of pepper keeps the richness in check.

From the red-sauce column, puttanesca is a long-standing favourite at Supermaxi in Fitzroy North, while Eat'aliano by Pino offers all'amatriciana: guanciale, San Marzano tomatoes, pecorino and, controversially, onion, which many purists say was never part of the original recipe. But then an amatriciana pizza couldn't be further from tradition, could it?

Connie's, Thurs-Sat nights, delivery up to 10km from CBD, conniespizza.com.au

Capitano, 7 nights, pick-up from Carlton or local delivery, capitano.com.au

Freddy's Pizza, Tues-Sun nights, pick-up from Windsor or delivery via Uber Eats, freddyspizza.com.au

Baby, 7 nights, pick-up from Richmond or delivery within 5km, babypizza.com.au

Ganzo, Wed-Sun nights, pick-up from Richmond or delivery to select suburbs, ganzo.com.au

Supermaxi, Tues-Sun nights, pick-up from Fitzroy North or delivery within 5km, supermaxi.com.au

Eat'aliano by Pino, pick-up from Windsor or local delivery, eatalianobypino.com.au