An amino acid was first isolated in 1806, by French chemists Louis Nicolas Vauquelin and Pierre Jean Robiquet
The proteins present in food are all composed of a group of amino-acids. The amino-acids constitute the building blocks for many different chemicals made inside the body: in hormones, enzymes, DNA. encoding, RNA. encoding.... they play vital roles in the body's immune system, they constitute the neurotransmitters in the brain, the cortisol which is involved with stress response - in short we know a fair bit about their roles in the body, and the body and mind can't function at full capacity without them.
About 500 amino-acids are known. Ref(1)
Altogether there are twenty amino-acids which we need. Of those twenty there are ten which the human body can synthesize, the other ten are only able to be obtained through our diet, these ten are called "essential amino-acids".
Animal proteins contain the amino-acids we can't produce ourselves, these are called complete proteins.
A vegetarian diet would still contain eggs and dairy products which, together with the vegetable proteins would cover the full count of the required ten essential amino-acids.
The question boils down to: can a vegan diet provide all ten essential amino-acids, if not, what supliments are available? (Which are not of an animal source)
It appears to be the case that vegetable proteins are on the whole likely to contain these amino acids, but some of them may be present in only small quantities. It is therefore necessary to find a variety of vegetable sources in order to obtain a balanced diet of complete protein.
From: vrg.org/nutrition/protein.php
Soybeans, quinoa (a grain), and spinach also are considered high
quality protein. Other protein sources of non-animal origin usually
have all of the essential amino acids, but the amounts of one or two
of these amino acids may be low. For example, grains are lower in
lysine (an essential amino acid) and legumes are lower in methionine
(another essential amino acid) than those protein sources designated
as high quality protein. Frances Moore Lappe, in her book Diet for a Small Planet
advocated the combining of a food low in one amino acid
with another food containing large amounts of that amino acid. This
got to be a very complicated process, with each meal having specific
amounts of certain foods in order to be certain of getting a favorable
amino acid mix. Many people got discouraged with the complexity of
this approach.
It may not always be possible or convenient to eat healthily on a vegan diet, but there are plenty of food supplements containing the individual elements you may need for that hard to get complete diet. Using a search engine to seek: "vegan amino-acids" many websites offer individual amino acids and quite a few pharmacies/health and fitness stores in your local town or city probably do the same. They do in mine. It would then be advisable for the individual to ensure that they are buying not only the appropriate supplements, but ones not obtained from animals.
Anyone embarking on a vegan diet would be well advised to seek advice from a dietitian about their specific requirements, particularly regarding feeding expectant mothers, children, young adults, the old and the very active or ill.
Refs:
1: Wagner, Ingrid; Musso, Hans (November 1983). "New Naturally Occurring Amino Acids". Angew. Chem. Int. Ed. Engl. 22 (22): 816–828.
Edit (1) - to include broccoli and a useful resource: Although the specific statement in the question refers to the website: VeganStreet.com, this answerer is unable to find the specific page making the claim, however there are other sites which make similar claims. Broccoli is not considered a complete source of essential amino acids, see here. A serving of 91 grams of the raw product contains:
> Tryptophan 30.0 mg
> Threonine 80.1 mg
> Isoleucine 71.9 mg
> Leucine 117 mg
> Lysine 123 mg
> Methionine 34.6 mg
> Cystine 25.5 mg
> Phenylalanine 106 mg
> Tyrosine 45.5 mg
> Valine 114 mg
> Arginine 174 mg
> Histidine 53.7 mg
> Alanine 94.6 mg
> Aspartic acid 296 mg
> Glutamic acid 493 mg
> Glycine 81.0 mg
> Proline 100 mg
> Serine 110 mg
>
> Attribution:
> http://nutritiondata.self.com/facts/vegetables-and-vegetable-products/2356/2#ixzz30ZrrGvhy
The bio-availability of these amino acids is not stated in the raw product, nor the cooked, nor is the RDA of any of them listed here.
The claims on the above site regarding the relative protein contents of broccoli and beef (per amount measured arbitrarily in calorific content) are contradicted here with citations, notably of USDA’s Agricultural Research Service’s Nutrient Data Laboratory database. It gives a list of the essential amino acids and their RDA. taken from the WHO database.
For reasons of complex formatting I have not quoted this, you should view the list here:
http://eathropology.com/2013/04/08/broccoli-has-more-protein-than-steak-and-other-crap/
It is clear that to ascertain the minimum daily requirements of for example histidine, approximately 20 cups of broccoli need to be consumed per day. The story is similar for other essential amino acids. To most people this would make broccoli an unfeasible source of all necessary dietary protein.