Place the butter in a mixing bowl. Pour in the walnut oil, honey, and the kosher salt.
Beat, starting on low speed until the butter starts to blend into the oil and honey, about 30 seconds. Raise the speed to medium and continue beating until the butter is smooth and creamy, about 1 minute.
Toss in the toasted walnuts and blend in on low speed. Scrape down the sides of the bowl to make sure all the butter is blended into the walnut oil and honey.
Let the butter chill for about 2 to 3 hours to allow it to set up slightly, making it easier to handle.
Transfer the butter to a serving dish or scoop it onto a sheet of food film and shape it into a roll.
Let the butter chill to completely firm up and serve chilled. The butter will soften as it warms – keep refrigerated for several weeks or frozen up to 6 months.