In a large bowl, combine the warm milk, yeast, and sugar. Let sit for 5–10 minutes until foamy. Add the flour, salt, eggs, and softened butter. Mix until a shaggy dough forms, then knead for 8–10 minutes until smooth and elastic. Kneading to a smooth, elastic consistency develops the gluten structure needed to trap gas evenly during rising, which is what gives the finished roll a soft, open crumb rather than a dense one.

Transfer the dough to a lightly greased bowl, cover, and let rise until doubled, about 1 hour to 1 hour 15 minutes. Letting the dough genuinely double, rather than cutting the rise short, gives the yeast enough time to build the volume of gas needed for a properly light, soft crumb once baked.

Turn the dough out and divide it into 4 equal portions. Shape each into a tight log, about 18–20cm long, and place on a parchment-lined tray. Cover and let rise again for 45–60 minutes, until puffy. Shaping the dough into a tight, even log gives the finished roll a cohesive crumb structure that holds together well once it's split, lightly toasted, and loaded with hot filling.

Bake in a preheated 190°C oven for 18–22 minutes, until golden-brown. Let cool before slicing. Do not overbake. You want a soft hero-style roll with enough structure to hold the filling. Pulling the rolls once they're properly golden, rather than baking further for extra colour, preserves the specifically soft crumb this style of roll needs — an overbaked roll trends toward a thicker, chewier crust that works against the sandwich's intended soft, bodega-style bite.

Dice the tomatoes, toss with the 2g salt, and let them sit in a sieve or bowl for 10–15 minutes so excess liquid drains off. This concentrates the tomato flavor and keeps the sandwich from getting watery. Mix the mayonnaise with sriracha or hot sauce if using. If you want it classic, keep the mayo plain. Salting the tomato draws its internal liquid out through osmosis before it ever reaches the sandwich — skipping this step means that same liquid releases later instead, directly onto the roll and into the warm beef filling.

Heat a large skillet or flat pan over medium heat. Add a small amount of oil if needed, then add the diced white onion. Cook for 8–10 minutes, stirring often, until softened, golden, and lightly caramelized. Remove to one side of the pan or to a small bowl. The onion should be sweet and lightly browned, not raw and not deeply dark. This is a lighter caramelisation than a fully slow-cooked jammy onion — just enough time and heat to soften the onion and develop real sweetness before it's chopped through the beef, without needing the much longer cook a deeply caramelised onion preparation would require.

Increase the heat to medium-high. Add the ground beef in 4 separate portions so you can build each sandwich more easily, or cook all at once and divide later. Season with the salt, black pepper, garlic powder, and onion powder. Let the beef sear first, then start breaking it up. Add the caramelized onion into the beef and keep chopping and mixing so the onion distributes evenly through the meat. Cook until the beef is browned, juicy, and lightly crisped in places, with no grey wet patches left. Letting the beef sear before breaking it apart, rather than crumbling it immediately, is what produces real browned flavour — beef broken up from the moment it hits the pan releases moisture too quickly and tends toward pale, steamed crumbles instead.

Pile the beef into 4 portions in the pan. Place 2 slices of American cheese over each portion. Let them soften for 20–30 seconds, then chop and fold the cheese into the beef until the mixture becomes cohesive, creamy, and classically cheesy. Do not overcook once the cheese is in. The filling should stay juicy, not tight and greasy. This chop-and-fold technique is the specific move that defines a chopped cheese as distinct from any other cheeseburger-adjacent sandwich — actively working the melting cheese directly through the beef, rather than letting a slice simply melt on top, is what produces the sandwich's signature creamy, cohesive texture throughout the filling rather than a separate cheese layer.

Slice the hoagie rolls open. Lightly toast the cut sides on a dry pan or with a touch of butter until just warm and lightly golden. Spread mayo on both sides of each roll. Toasting only lightly, stopping well short of a fully crisp exterior, protects the sandwich's specific soft-roll eating experience — too much crisping and it stops reading as a genuine chopped cheese and starts feeling like a different, harder-textured sandwich entirely.

Fill each roll with one portion of the hot chopped beef and cheese mixture. Top with shredded lettuce and the drained diced tomato. Close the sandwich gently and, if you like, wrap it loosely for 1–2 minutes so everything settles together slightly, just like a bodega-style sandwich. That brief wrapped rest lets the residual heat from the beef gently warm and slightly wilt the edges of the lettuce and soften the assembly into a single cohesive sandwich, mimicking how a chopped cheese comes together when wrapped in foil or paper at an actual bodega counter.