Off the top of my head... This is how I'd handle it...
The elipsis DOES signal the reader that the dialogue is ongoing... You can also use the elipsis with action to do the same thing.
No need for CONTINUOUS or (CONT'D). Less is more with specs which are meant to be read. If there's NO REASON to slow down the read? Don't do it. Readers KNOW the scene is continuing without CONTINUOUS. Readers know the dialogue is continuing without (CONT'D).
I just assumed this was a spec... If it's to actually shoot? Write it any way you want.
Also? You ORPHANED the INT. KITCHEN - CONTINUOUS by placing dialogue right after the scene location heading. Never orphan a scene location heading.
Last but not least? I used secondary scene location headings instead of master scene location headings as what you wrote didn't FEEL like the very beginning of the script.
*NOTE: The only time I DO USE (CONT'D) is when the software splits the dialogue at the bottom of a page and begins on the new page.
*NOTE: Oops. Forgot the "l" in talk. LOL. I did this very fast.