A two-stage, slow-start valve includes a minor valve assembly inside a major valve closure member of a major valve closure assembly so that fluid can flow through the valve at a higher rate by opening the major valve assembly or at a lower rate by opening the minor valve assembly while the major valve closure assembly is closed in order to allow fluid to flow in an alternate flow path through the major valve closure member. The example minor valve closure assembly has a minor valve closure member that is actuated to open by a minor valve piston on which minor valve actuating fluid is applied to the minor piston via a diaphragm seal positioned between the minor piston and the minor valve actuating fluid. Both the major valve closure assembly and the minor valve closure assembly can be pneumatically actuated to open.