Something in the fuel tank
or in the carburator could cause this behavior. So might a worn (too flexible) vacuum hose. The rest period lets the object (dirt?) fall to the bottom of the tank or the bowl, (or hose to return to shape) allowing you to startup. At about 30 min. the suction of the fuel (or the heat of the engine with vacuum lines) moves the obstruction blocking the fuel line, carburator jets, or collapses the weak hose.
Start by inspecting the vacuum hoses when the stall occurs, OR disassembling the easiest one to work on. Inspecting and cleaning the item, check that the carburator float does not have a pinhole filling it and sinking causing flooding.