Season the tuna with the salt and pepper, and place the wood chips in the bottom of a stovetop smoker
and position the drip tray and wire rack over the top according to the manufacturer's instructions. Lightly oil the wire rack, then place the tuna slices on the rack and position the lid over the top of the smoker, leaving slightly ajar. Place the smoker over high heat until smoke begins to exit the smoker. Reposition the lid so that the smoker is completely covered and lower the heat to medium. Cook the tuna until just cooked through, about 8 minutes.
Chop the tuna into 1-inch pieces and place in a food processor with the cream cheese, butter, green onions, shallots and parsley. Process until a very smooth paste, stopping to scrape down the sides of the bowl as necessary. Add the heavy cream and pulse until just combined. Transfer spread to a small serving bowl and serve the spread with crisped bread or spread on crackers, garnished with green onions or parsley, as desired.