Place the peeled, cubed potato in a small saucepan and cover with cold water. Bring to a boil over medium-high heat, then reduce to a simmer and cook until very tender, 10–12 minutes. Drain well and return the potato to the warm pot. Mash until completely smooth with no lumps. Measure out 1/2 cup of mashed potato and let it cool to barely warm or room temperature.

In a large mixing bowl, beat the softened butter with the mashed potato, vanilla, and salt until smooth. A handheld mixer works well here.

Add the powdered sugar 1 cup at a time, mixing on low speed after each addition until incorporated. The mixture will become quite thick. Stir in the shredded coconut with a sturdy spatula or wooden spoon until evenly combined and no dry coconut remains. The mixture should be thick and scoopable, like stiff cookie dough. If it’s very soft, add more powdered sugar 1–2 tablespoons at a time. If it’s crumbly and won’t press together, sprinkle in a teaspoon or two of milk or water.

Line a 9x13-inch baking pan with parchment paper, leaving some overhang for easy lifting. Press the coconut-potato mixture evenly into the pan, smoothing the top with the back of a spatula or your hands. Chill in the refrigerator for at least 1–2 hours, or until very firm, or in the freezer for 30–40 minutes. Once firm, lift the slab out of the pan using the parchment and place it on a cutting board. Use a sharp knife to cut small squares or rectangles about 1–1½ inches each. Return the cut pieces to the fridge while you prepare the chocolate.

Place the chocolate and coconut oil or shortening in a microwave-safe bowl. Heat in 20–30 second bursts, stirring well after each, until smooth and fully melted. Alternatively, melt in a heatproof bowl set over a pan of barely simmering water, making sure no water gets into the chocolate. The chocolate should be fluid but not scorching hot.

Line a baking sheet with parchment or waxed paper. Working with a few pieces at a time, use a fork to lower each chilled coconut square into the melted chocolate, spooning chocolate over the top to cover completely. Lift the piece out, gently tap the fork on the side of the bowl to remove excess chocolate, and slide the candy onto the lined baking sheet. Repeat with the remaining pieces, reheating the chocolate gently if it begins to thicken. If desired, drizzle a little extra chocolate over the tops or sprinkle with coconut before the coating sets.

Refrigerate the dipped Needhams for 30–45 minutes, or until the chocolate coating is firm. Trim any excess chocolate “feet” with a small knife if cleaner edges are desired. Store chilled until ready to serve.