Here is a list of vegetables that can be planted for June 1st according to the Rhode Island Planting Calendar produce by the URI Master Gardener Association.
From seeds (plant directly in the garden):
Baby lima beans
NOTES: Dates are based on final frost on May 15 and first frost on October 15.
Be aware of local microclimates that may make your garden cooler or warmer & alter frost dates.
Transplants should be exposed to the outdoors for increasing periods over several days before planting.("hardened off")
Download the complete calendar here