USCIS will begin premium processing for all remaining FY2020 H-1B cap-subject petitions on 6/10/19. Petitioners may file a Request for Premium Processing Service, with the USCIS service center processing their petition on that date.
More information can be found at https://www.uscis.gov/news/alerts/premium-processing-begins-remaining-h-1b-cap-subject-petitions-june-10.
Ann Massey Badmus 