A 'heavy' breakfast is commonly used in AmE.
It's somewhat subjective to the individual speaker, but for me 'heavy' means loaded with carbs and/or sugar. It sits 'heavy'* in your stomach - the kind of stuff that makes you a bit drowsy or lethargic after eating. It's the kind of breakfast that fills you up so much you end up skipping lunch later.
It does have negative connotations as opposed to 'big' or 'hearty'. Those are simply large meals, but they don't necessarily sit heavy* in your stomach.
For example, I don't eat pancakes (or waffles) for breakfast, because to me they're too 'heavy'. I generally can't handle sugary foods (including donuts) for breakfast.
I just recently asked a colleague if he wanted to grab an early lunch before our meeting, and his reply was "No thanks, I had kind of a heavy breakfast. I'll eat after the meeting."
*Yes, grammatically it should be 'sit heavily', with an adverb, but the set expression (at least in AmE) is 'sit heavy'