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."
There are two aspects in your question:

1. the quantification over sandwich
2. the quantification over time

> 
> « J'ai mangé beaucoup de sandwichs que ma mère a préparés. »   
> 
>  => “I ate lots of sandwiches made by my mum.”
> 
> 
> 

This says nothing about sandwiches made by mum I did not eat and nothing either about how many times mum prepared sandwiches. I might have eaten lots of mum-made sandwiches over time.

> 
> « J'ai mangé beaucoup des sandwichs que ma mère a préparés »  
> 
>  => I ate a lot of the sandwiches my mum made.
> 
> 
> 

This says I did not eat all the sandwiches made by mum and also seems to imply that we are talking about one lot of sandwiches mum made that one time.

The first sentence is a sort of indefinite, the second is a sort of partitive.