Yes, do make something out of your branch! We’ve made small spoons out of smaller wood than that. It will be a lighter wood (color and density) than a larger piece with a lot of heartwood but it will still make something usable. Green wood is easier to carve than dry but it needs a bit of care so it doesn’t split when drying (fine for a spoon you’d be getting thin soon). Dried is better for anything that will have more thickness (like a tool handle) . Walnut is nice wood to carve. If you want to let it dry first then leave the piece as long in length as you can (it will likely split in from the ends), bark on, out of the sun, maybe paint the ends with something if it’s a small piece, and just let it dry naturally for a year or two. Then split it in half and make something out of those pieces (avoiding the center pith). A tool handle or simple shallow spoon or stirrer maybe. If you don’t want to wait, make a spoon out of the green wood (many good books and articles around about spoon making). It’ll have meaning when you use it and so give double joy!
We’ve carved spoons for forty years and used all types of wood, including basswood, poplar, pine and autumn olive. These aren’t first choices, of course (we’ve used mostly hardwood) but they’ve made good, long lasting spoons. Your walnut will work just fine. Sue
PS - As an aside, for interested carvers looking for ideas, you might check out Norman Stevens “A Gathering of Spoons” - an amazing collection of what different folks can do with a humble small piece of wood and a ‘simple’ spoon!