Lamb's ear: Either. Will be a mushy mess in the spring - yucky, but very easy to pull/yank out, not much exertion required for clean-up at that stage.

Catmint: Same - in fall after die-back or in spring. Can also shear mid-summer to keep in bounds and encourage more flowering - responds beautifully to this treatment.

Both are very low-care plants :0)

Whatever you leave will become mulch. On both of those plants, the stems are not woody at all and decompose quickly enough that they settle down under new spring growth, which completely covers it.

If the lambs ear is the blooming kind (some types are bred not to bloom) the dead flower stalks likely contain a lot of seed. The same is true for catmint. One way I like to propagate both those plants is to shear the stems with some heavy-duty kitchen scissors (or anything else that works) and toss the dead stuff somewhere I want them to grow. Both types are vigorous re-seeding plants that will go from seedling to blooming plant by next summer if they germinate in the fall.

Be careful about composting the stuff because then your compost will erupt into blankets of plants wherever you put it. That's one reason to leave the dead material in place where it can benefit the garden and only multiply in place.

Thanks all!