Broken buttercream looks catastrophic but is almost always recoverable if you understand what caused it. The problem is nearly always temperature, not technique. Buttercream is an emulsion, and like all emulsions, it depends on the fat and liquid being at compatible temperatures to stay combined.
Swiss meringue buttercream splits most often when cold butter is added to a meringue that has not cooled sufficiently. The butter melts partially on contact, releasing fat that cannot re-emulsify at that temperature. The mixture looks soupy or curdled. The solution is to keep mixing. As the butter cools the meringue further and the temperature equalises, the emulsion usually comes back together within 3 to 5 minutes.
When the room temperature is the actual variable
In warm kitchens above 24 degrees Celsius, even properly tempered butter softens faster than the meringue can absorb it. The result is a greasy, separated mass that does not recover easily from continued mixing because the fat has already pooled. Chilling the bowl briefly in the fridge for 10 minutes, then returning to the mixer, brings the temperature back into range.
American buttercream, which contains no meringue, breaks differently. Too much liquid added too fast overwhelms the fat-to-sugar ratio. Adding liquid one teaspoon at a time and mixing fully between additions prevents this entirely.