Line a large baking sheet with parchment paper. Place the pound cake cubes on the lined baking sheet in a single layer and bake until toasted and firm, about 10 minutes. Let cool completely.
Brush the toasted cubes with the butter. Mix the sugar and cinnamon together in a large bowl. Toss the cubes in the cinnamon-sugar. Thread the cubes on ten 6-inch skewers. Fill each of 10 shot glasses a third of the way full with the dulce de leche. Stand one churro skewer upright in each glass of dulce de leche and serve.