I'll elaborate on Joe's answer some. It's a little known fact that button mushrooms (also called common, white, or table mushrooms), cremini (crimini, baby portobello, baby bella, brown mushroom), and portbello (portabella) are all the same species. Specifically, agaricus bisporus. The difference in appearance and taste is based solely on maturity. The youngest are white button mushrooms, and the oldest are the big portobello mushrooms. There can be slight differences in skin color for the portobello's, though it tends to be darker than the stark white of the button.
The mushrooms in that video are most certainly fully mature agaricus bisporus, a portobello.
It's a safe bet that anytime you see a mushroom being substituted for a "steak", that it will be a portobello. They have a rich meaty flavor and texture that is perfectly suited for this.
I've looked through a number of recipes, and the vast majority of them indicate that you should cook the morels thoroughly (preferably in butter) first, before using them as a pizza topping or in an omelet.
I did find at least one recipe for a morel omelet that did not call for pre-cooking the mushrooms, however.
I must admit that I did my research a bit late, as I had already decided to risk it and made an omelet with morels, as I had not yet received any answers to my question.
I added the morels to the raw egg, rather than putting them in as a last-minute filling just prior to folding the omelet, so they did cook a bit during the process. The result was actually quite delicious (I also used some fresh garlic, chives, and Gruyere cheese), and I did not become ill.
However, it does seem that some people react differently, and alcohol consumption can greatly enhance any negative reactions to wild mushrooms.
Given the information I found on the Internet (none of it terribly authoritative, as the vast majority of quality information was on small sites maintained by wild mushroom hunting clubs and enthusiasts), I would not repeat my omelet without pre-cooking the mushrooms, nor will I make pizza without pre-cooking them.
The consensus seems to be that they should be sauteed once until "thoroughly cooked". At least one site (amid heavy cautions) specifies that this is usually 8-10 minutes, although I suspect that this is for intact mushroom halves, rather than small pieces (morels should always be sliced lengthwise to ensure that you can remove all the insects from the hollow center).
It needs to be stressed that even with the proper cooking, it seems some people have strong reactions to wild mushrooms, so use caution and moderation when trying them for the first time, and do not try them with any form of alcohol until you are sure that you can eat them without problems.
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A major reason produce can be sold loose without tongs is that you're supposed to wash it just before preparing it anyway. You couldn't wash your baked goods. How effective this washing is, is of course another matter now we don't have to wash the mud off and a token rinse is probably typical.
Some produce, including mushrooms but also strawberries etc., could easily be damaged by tongs anyway.