Once you have received your new contract and are happy with it, I would suggest having a chat with your old Boss advising them that you have received an offer and are considering it.
It is more courteous for one, rather than slapping down your notice and could lead to a counter offer being proposed. Chances are, you are leaving anyway, but do it right and do not burn bridges. Those contacts could serve you well some day.
As someone who has worked in HR and the reruitment industry for 20 years, and as someone who employs people, this is the way forward, and what I would advise. You are essentially giving your current employer the option to try and keep you, which they will appreciate, rather than just telling them you’re off. If you really don’t want to stay, thank them for any counter offer they may make, and proceed with your written notice.