This is possible but will require basic knowledge of HTML, CSS and JS.
Lay Theme has provided options for users to be able to implement their own custom code with "Lay Options - Custom HTML & CSS"
Here you can add the necessary code for creating the burger menu you desire,
Also some reference for you if implementing your code:
The Basic concept behind menu's like this is that the menu already exists but outside the boundaries of the viewport, when the menu is 'clicked'
A class is added to the menu:
which gives it new CSS properties pulling it into the viewport,
The smooth sliding that occurs is from the CSS transition property:
The Example you like above also uses this method -
Here we have the menu before:
And now when clicked, the Button gets a class called "header__nav-cta--active" this adds new CSS for the arrow to face north
And also the Navigation gets a class added called "header__nav-wrap--open" Which adds new CSS making the hiding menu now visible
Extra for you :)
transition: all 0.8s;
-100% ( 100% off the right side of viewport)
is now 100% ( full width of viewport )
I hope this helps you Neil, I'm sorry that i can't pursue this further for you but i do wish you good Luck! :)
Also seeking the aid of a developer for a singular task like this could be an option for you,
Let me know if you have any further questions and best for your Lay Theme Journey.