Place the chopped unsweetened chocolate and vegetable oil in a medium, microwave-safe bowl. Heat in 30-second intervals, stirring between each, until the chocolate is completely melted and smooth. Set aside to cool slightly.
In a large mixing bowl, combine the granulated sugar, eggs, and vanilla extract. Use an electric mixer on medium speed to beat the mixture until it becomes pale and thick, about 2-3 minutes.
Pour the slightly cooled chocolate mixture into the egg and sugar mixture and stir with a spatula until just combined. In a separate medium bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing just until no streaks of flour remain. The dough will be soft and sticky.
Cover the bowl with plastic wrap and chill the dough in the refrigerator for at least 4 hours, or until it is very firm and easy to handle.
Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper. Place the powdered sugar in a shallow bowl. Scoop out about 1 1/2 tablespoons of dough, roll it into a ball between your palms, then roll it heavily in the powdered sugar until fully and thickly coated.
Place the coated dough balls about 2 inches apart on the prepared baking sheets. Bake for 10-12 minutes, or until the cookies have spread and the tops are crackled. The edges should feel set while the centers remain soft.