Lovely spoon! I have just looked up desert willow, and found that it is not a willow as I know it (salix), but called willow because the leaves are willow like. What does it carve like? Is it hard or soft? Salix over here is really soft, marks easily and is prone to fibres tearing out unless extremely sharp tools are used.
One source notes it makes good bows and judging by the grain in the second photo (being a carpenter & joiner) I would hazard a guess that it is much like Ash in workability green or dry with similar traits of flexibility.
It is very "fibrous," but to me that means the stuff doesn't split easily. It holds together something like elm maybe. I don't like the fact that it marks easily while working on it. I had to go over the whole thing again at the end with clean hands, carefully carving away the smudge marks.