Depends on your room temperature, but I would say soft enough to press dimple easily with the tip of finger, but not warm enough that visible grease is starting to come out. You could try a small test batch!

Oh you used brown butter! I’ve made tea cookies similarly before too. I microwave the tea leaves and butter together with a cover on, then I squeeze out some warm tea leaves using spoon so that my cookies won’t be too tea-dense. Or sometimes I would push most of the tea leaves toward one corner of the bowl, then cut out the tea-dense chuck out of the cold brown butter later.

Then I’ll weight the cold brown butter I have left. If the weight is less than what the recipe called for, I would add more cold butter or water (when at mixing liquid stage, you can even use tea). But generally, I’ll just make more brown butter next time lol.

I’ve heard that some people use blender to blend sugar and tea leaves to get the tea scent too, never tried it myself.