I push slices a lot in my lectures on development. Whether it's for determining a minimum viable product, or breaking things down into testable releases, you can't be too thin. Elephant carpaccio as Cockburn says.
But slicing what you build is not the only place to look. Sometimes, it's important to think about slicing your target users. I was just hearing about a project rollout in crisis in a large enterprise because of the number and variety of users who need to be updated. It's basically your classic updating of the base level enterprise operating systems and all the attendant applications that affects.
Mass rollouts are famous for, if not failure, at least a lot of grief and midnight oil expenditures. If it hurts when you do that, don't do that! Slice the rollout. But just as you have to think about how to slice a product, so that you don't slice out the important part of the product, so too you need to think about how to slice your initial targets. Doing it division by division or region by region is as bad if not worse than a mass rollout. The first division in line gets all the bugs and missteps, often becomes disconnected from the rest of the company, and rightly feels like a lab rat.
Instead, look for the small audience that would be OK with glitches in the change process. What group really really wants this change, or what group is pretty tech-savvy and hence likely to find workarounds give useful reports on what's broken?
Don't stop with one slice. "We had a pilot rollout, now we're going live everywhere." Pilot groups are never representative of the total audience. Even if your initial slice was not a tech-savvy group, they knew they were in an experiment and almost certainly gave you the benefit of the doubt. That's not going to be true of everyone else.
Instead, you need to keep slicing. There are many criteria for the next slice. It might be the next group who most wants the change. It might be the group your first group interacts with the most. It might be the group that's most concerned about the change. Why that last group? Because at this point you can still afford to give them the extra attention and handholding they need at this. Plus you find out what the worst case reactions and snafus are.