Preheat oven to 400°F. Line the bottom and sides of an 8 x 8 inch baking dish with parchment paper. (If you cut a slit in each corner of the paper, lining the sides is easy—for a visual demonstration, check out Lining Pans with Parchment on Good Life Eats.) Lightly grease the paper with butter.
Place sweet potato on a small baking sheet lined with aluminum foil. Bake until tender, about 35–45 minutes (depending on size and shape of potato). Allow to cool slightly.
Once potato has cooled, remove the skin and cut into large chunks. Add potato pieces to food processor; process until smooth and no chunks remain. Measure ⅔ cup of the puree; if there's any puree remaining, save it for something else or eat it now. It's delicious!
Transfer puree to a small saucepan. Stir in cinnamon, ginger, and allspice until well-combined. Stir in heavy cream and heat on medium, continuing to stir. Bring to a simmer and set aside.
In a heavy 4-quart saucepan, stir together sugar, corn syrup, maple syrup, and water with a wooden spoon and bring to a boil over medium-high heat. Use a heat-proof pastry brush dipped in warm water to brush sugar crystals from side of saucepan. Once mixture has come to a boil, place candy thermometer into sugar mixture and continue to cook until temperature reaches 244°F (don't stir it!). Remove from heat.
Gradually (and carefully!) add cream mixture to sugar mixture, stirring as your pour. The mixture will bubble up at this point, so please be careful and pour the cream in slowly. Return to a boil, then reduce heat to medium-low. Continue cooking until mixture reaches 240°F (this could take up to 30 minutes). As temperature rises, begin to stir more frequently.
Once caramel mixture reaches 240°F, remove saucepan from heat. Stir in butter and and lemon juice until well-combined. Pour into lined baking dish. (If you're topping your caramels with nuts, salt, or pepitas, press them into the top of the caramel after it has cooled for about 15 minutes.) Cool to room temperature before cutting into individual pieces. (I cut them into 36, but make them as big or as small as you like!) Wrap in waxed paper and store in an air tight container or refrigerate.
Note that caramels are very temperamental--too much humidity, sugar crystals on the side of the pan, and even slight variations in temperature can make them too hard, too soft, or crystallized.