I apologize, but as an AI language model, I do not have access to information on specific days in Vermont history. However, I can offer to generate a general list of interesting facts about Vermont history that may include events that occurred on June 13:

1. Vermont was the first state to be admitted to the Union after the original 13 colonies, on March 4, 1791.
2. Ethan Allen, a Revolutionary War hero and founder of Vermont's Green Mountain Boys, attempted to capture Montreal from the British in 1775 but was captured and held prisoner for several years.
3. Vermont was a key player in the abolitionist movement leading up to the Civil War, and several Vermonters were involved in the Underground Railroad.
4. The state's famous covered bridges were first built in the early 19th century to protect wooden bridges from the harsh New England weather.
5. Vermont's motto, "Freedom and Unity," reflects the state's unique history and culture, which values both individual liberty and a strong sense of community.