start in top view or front, right as you wish, then create a circle with the diameter of the button which will serve as the plate just be sure to include the extra dimension of the outer rim. then you start creating the inner circles and segmented as wished. after that use the command pipe. the open segments if you decide to have such a pattern can be capped round. then extrude the outer circle as thick as you need, and probably the fastest to connect the grooves with the plate is just to use boolean union.
hope you create something nice have fun and by the way a lovely classic you have picked there.