Vegetarians: When you go to a restaurant, how can you be sure that you aren't consuming any animal product?

I require a written statement confirming the full and legitimate veganity of said dishes, signed by the manager of the restaurant, the head chef, and the director of the company, before I agree to peruse the menu. 

You know because you ask. All of the restaurants I've managed and worked in, had no issues in putting vegan/vegetarian meals together in clean pans with non animal fats or butter. No two meals were cooked in the same pans unless the meals were going to the same table and not even then if different sauces or seasonings were used.

Cross contamination was avoided between, grilled vegetables, meat, fish and chicken, by using grill pans for things that needed to be separate from other foods. 

For those stores that had limited equipment, the guest were informed how we could or couldn't prepare things, and let them make up their minds to oven roast, steam or boil vs using the flat top or limited grill space, unless it was slow enough to do a decent cleaning (normally between shifts). Most fryers had vegetable oil/shortening vs beef or pork fat/shortening or lard. 

Vegetarians eat animal products. They just don't eat MEAT of land animals or birds. They definite consume eggs, cheese, milk, honey, etc.  Many even eat seafood.

You are confusing vegetarians with vegans.  

Vegans eat no animals and nothing that COMES from animals. 

And there is no guarantee of what you are eating unless you cook it from scratch yourself.  

Going vegetarian or vegan is a choice, and eating animal products is not harmful.  In fact, as many vegans and vegetarians age, they have to start eating meat to retain their health. Even the Dalai Lama has been ordered by his medical doctors to eat meat.

My own sister - a vegetarian nutritionist before she retired - was a vegetarian for over 35 years, but eventually had to start eating meat again so that she could retain her energy and strength.  But it is FINE to eat vegan/vegetarian as long as your health remains strong.

You can't be sure of anything when someone else is handling your food.

I don't think you really understand what a vegetarian is. Dairy products, such as butter is part of a vegetarian diet. The majority of vegetarians simply avoid eating meat or anything that comes from a dead animal. They're not obsessive, crazy people! Also, in many countries, if something is claimed to be “vegetarian” then that's what it has to be. Obviously some people break the law or or ignorant of regulations but most of us just accept that things are what they are claimed to be.

I'd guess you wouldn't want to eat rat or cat, but how could you be SURE it wasn't included in something you ate? You can't be, you just accept that it isn't. Otherwise you'd struggle to find food.

Who cares? The point is to support a vegetarian option with your money. That's what counts in the end, not avoiding every little bit of fat.