#1 is Simple Past - it boils down to ‘The assistant started.’
Present Progressive is formed by taking the present tense of ‘to be’ plus the present participle of the main verb
‘The assistant is starting’
#2 is again Simple Past. ‘The assistant started’
Present Perfect Progressive is formed by using the present tense of ‘to have’ plus the past participle of ‘to be’ plus the present participle of the main verb.
Both of your sentences use participles, but participles are just verb forms, not tenses. Both sentences are entirely in the past.
Sentence 1 could be reworded to begin this way:
After she cut the vegetables and laid … (past simple tense)
OR
After she had cut the vegetables and had laid … (past perfect tense)
You don’t actually need the past perfect in the reworded version of the sentence because the word “after” makes it clear that the meat preparation came later.
Sentence 2 uses the perfect participle form: “having cut”. For the sake of consistency, it would be better to use “laid” rather than “laying” in sentence 2:
…, having cut the vegetables and (having) laid them on the table, …
Both “having cut” and “having laid” indicate that those actions were already complete before another action (i.e. before “started to prepare the meat”). Since your second sentence does not include the word “after”, the use of the perfect participles takes care of that idea. Thus it is similar in meaning to the past perfect. You might reword sentence 2 this way:
The chef’s assistant, who had cut the vegetables and b laid[/b] them on the table, then started to prepare the meat.