Peel and cube a medium russet or Yukon Gold potato. Place in a small pot, cover with cold water by about 1 inch, and bring to a gentle boil. Cook until the pieces are very tender when pierced with a fork, about 10–12 minutes. Drain well and return the potato to the warm pot. Mash until completely smooth with no lumps (a ricer or pressing through a fine mesh sieve gives the smoothest texture). Let the mashed potato cool to room temperature before using so it doesn’t melt the butter.

In a large mixing bowl, beat the softened butter with a hand mixer until creamy. Add 1/2 cup cooled mashed potato, vanilla, and salt; beat until smooth and well combined. Gradually add 4 cups of the powdered sugar, about 1 cup at a time, mixing on low speed. Stir in the shredded coconut with a spatula. Check the texture—you want a thick, scoopable dough that holds its shape when pressed, firm but not crumbly.

Add additional powdered sugar, 1/4 cup at a time, until the mixture is firm enough to press into a pan or shape into squares. If the mixture is too soft and sticky, add more powdered sugar. If it becomes too stiff or dry, sprinkle in a teaspoon or two of mashed potato or a teaspoon of milk, mixing gently and adding very slowly.

Line a 9×13-inch baking pan with parchment paper, letting it overhang the edges. Press the coconut-potato mixture evenly into the pan, smoothing the top with an offset spatula or the back of a spoon. Chill in the refrigerator for at least 2 hours, or in the freezer for 45–60 minutes, until very firm. The centers must be cold and solid before dipping so they don’t fall apart in the chocolate.

Lift the chilled slab out of the pan using the parchment overhang and place it on a cutting board. Using a sharp knife, cut into small squares or rectangles about 1 to 1 1/2 inches in size. You should get roughly 40–50 pieces, depending on how large you cut them. Return the cut centers to the fridge or freezer while you prepare the chocolate.

Place the chopped chocolate and coconut oil (or shortening) in a microwave-safe bowl. Microwave on 50% power in 20–30 second bursts, stirring well after each. Stop heating when most of the chocolate is melted with a few small pieces remaining; continue stirring until the mixture is completely smooth and glossy. If using a stovetop, melt gently in a double boiler over barely simmering water, stirring until smooth. If the chocolate seems very thick, stir in another teaspoon or two of coconut oil to thin it slightly for dipping.

Working with a few chilled centers at a time, place one piece on a fork and dip it into the melted chocolate, spooning chocolate over the top if needed. Lift it out and gently tap the fork on the side of the bowl to let excess chocolate drip off. Slide the coated candy onto a parchment-lined baking sheet. Repeat with the remaining centers, reheating the chocolate briefly if it begins to thicken. Optionally, drizzle extra chocolate over the tops or sprinkle a few coconut flakes on some pieces for decoration.

Let the dipped Needhams sit at room temperature until the chocolate coating is firm, about 30–45 minutes, or refrigerate for 15–20 minutes to speed setting. Once the chocolate is fully set, transfer the candies to an airtight container. Store at room temperature (in a cool room) for up to 3 days, in the refrigerator for up to 2 weeks, or freeze for up to 3 months with parchment between layers.