Brown the beef properly: heat a 12-inch skillet until genuinely hot. Add the beef in one layer, press it flat, and do not touch or salt it for 3-4 minutes - salt at this stage pulls water out and you poach the meat instead of browning it. Once a crust forms underneath, break it up and cook through to 160°F (71°C). If your largest pan is 10 inches, brown in two batches.
Pour off about half the fat: a pound of 85/15 renders roughly 1 1/2 tablespoons. Pour off about a tablespoon and leave the rest in the pan. Cook the onion and bell pepper in it 4-5 minutes until soft, then the garlic 1 minute more.
Bloom the tomato paste: push everything to one side, drop the tomato paste into the clear space and fry it 60-90 seconds, stirring, until it darkens from bright red to brick and smells sweet rather than sharp. Then stir it through the meat. This is the single largest flavour gain in the recipe and almost no low-carb version does it.
Add the liquids and reduce uncovered: stir in the broth, water, Worcestershire sauce, vinegar, mustard, molasses, sweetener, chilli powder, smoked paprika and pepper. Simmer uncovered 10-12 minutes, stirring occasionally, until a spoon dragged across the pan leaves a channel that closes slowly. A lid returns every drop of evaporating water; reduction is what thickens this, which is why there is no flour or gum in the list.
Taste last, salt last: add the salt a little at a time. The broth, Worcestershire and mustard all brought salt and the sauce has just concentrated by a third. If it tastes flat rather than salty, it usually wants another splash of vinegar instead.
Toast the buns: split and toast the cut faces in a dry pan or under the broiler until golden. Low-carb buns are drier and more porous than wheat and go soggy in about two minutes untoasted.