I have brewed with potatoes, and a friend of mine has brewed with carrots.
He merely put carrot juice into secondary fermentation. It wasn't a bad beer, by any stretch of the imagination and WAS a little orange. So juicing is an option.
If you think about it, making beer from a root vegetable shouldn't be that much different than making vodka from a root vegetable. In either case, you want to create a fermentable liquid. In your case, you're going to act upon with ale yeast instead of distiller's yeast, and you're not going to ferment it.
With potatoes, when you're making vodka, you generally use a pressure cooker to completely pulverize a potato in a small amount of water and then use that 'potato juice' as a basis for the fermentable liquid.
I did a little bit of research on potatoes before using them in my beer. While I plan on totally pulverizing them next time (to see how it comes out), I can tell you that the starches in potatoes gelatinize at about the same temperature as the starches in barley. I did a pre-mash with my potatoes at 150F for an hour before adding them to my normal mash. The upside to this is that you don't get a bunch of goopy potato in your mash. The downside is that you probably don't get as much starch/sugar out of them as you would in the pulverization method.
I am not convinced that baking a potato and adding it to your mash gets you anything more than a little bit of flavor (I'm sure you lose a lot during boil/fermentation).
If you really want a beer to taste like a vegetable, I would pasteurize it (or bake it, whatever) and add it to secondary fermentation, rather than the mash.
Here are my initial thoughts/research results when I first made my potato beer:
I'll be doing it again in a few weeks now that I had a little more knowledge and I'll be attempting to actually measure how much gravity I'm getting from the potatoes. I'll be sure to post the results.