Tilleys are pretty fantastic. They are also the reason all my fedoras look so good. If I am going to be sweating and doing hard work I wear my Tilley or pith helmet so they take the brunt of abuse.I love that. Tilleys are what I believe to be some of the most versatile hats out there. I bought an airflow LTM5 years ago for sun protection (still wear it) and now have six in different styles/colors. My latest one is a hemp in olive green that looks a lot like the one you're wearing.
Looks like your beavers are as ambitious as ours!