Cover the bottom of a big Dutch oven with wood chips. Ball up three big pieces of aluminum foil each to about the size of a golf ball, then place them on top of the chips; they'll be used to prop up the bowl of butter. Cover the Dutch oven and heat over medium-high heat for 10 minutes.
Place the butter in a heatproof bowl and place the bowl on top of the aluminum balls. Reduce the heat to medium-low and cover the pot, allowing a small opening to vent. Cook for 30 minutes. Taste; if you'd like it to be smokier, cook for longer, up to 30 minutes more. Let the butter cool at room temperature back to 65 to 70 degrees F.
To speed up the cooling, stick the butter in the freezer, stirring occasionally, until spreadable and opaque, about 20 minutes.
Sprinkle the butter with flaky salt, then serve with radishes for dipping or spreading.