Question: I have been told that you always use 'beaucoup de':

I understand the below should use 'de'

J'ai mangé beaucoup de sandwichs
(I ate lots of sandwiches)

However, I was told even when you are saying 'lots of **the** sandwiches', you still use 'de', which I am not satisfied with:

J'ai mangé beaucoup **des** sandwichs que ma mère a preparé
(I ate lots of **the** sandwiches that my mum made)

Should 'de' be used in the above or can you use 'des' in a case when you are referring to specific things?

Answer: 
You're right, it would make no sense to use *de* in that situation. "Beaucoup de sandwiches" means "many sandwiches", not "many of the sandwiches."

In my view, there is nothing wrong with "beaucoup des sandwiches que ma mère a préparés." (But note the agreement of *préparés* with its direct object *sandwiches*.) 

It may be that the person you spoke to feels that "beaucoup des sandwiches" is not acceptable, or not elegant, and didn't really understand what you were getting at, so they suggested "beaucoup de." An alternative would be "plusieurs des sandwiches que ma mère a préparés."
When you are referring to specific things, you use definite articles. In the case of a plural, it is "*les*".

Now, "*beaucoup de les*" is unacceptable, since "*de les*" always contracts as "*des*".

In your example, the sandwiches are not generic; they are those your mother prepared, so a specific thing. Your example is perfectly fine!

Now, when speaking of generic things, this would be another business:

> 
> I ate lots of sandwiches.  
> 
> *J'ai mangé beaucoup de sandwiches.*
> 
> 
> 

It is however the same logic as in English. Your instinct seems better to me than what you were told. I cannot think of any counter example at the moment.