Change the "Lost Your Password" process

Does anyone know if it is possible to change the process that a user goes through to get a new password?

Specifically:
– rather than WP sending users two emails (an email verification email and then another one with a new password), could…
– WP send a single email with a link that leads a user to a page that asks for a new password?

This kind of change still involves email verification (with the link), greatly simplifes the new password process (particularly for less-savvy users, like the many seniors in one of my sites), and removes the long/confusing system-generated passwords (although they are highly secure).

So does anyone know of a way?

Many thanks!