Heat oven to 350°F. Line 2 baking sheets with parchment. Whisk together the flour, baking powder, baking soda, and salt in a medium bowl.
Beat the butter and sugars in a large bowl with an electric mixer on medium-high until light and fluffy, 2 to 3 minutes. Beat in the egg, then the vanilla, scraping down the sides of the bowl as necessary. Reduce mixer speed to low and gradually add the flour mixture, mixing until just combined. Fold in ¾ cup of the chocolate chips.
Drop mounds of the dough (about 1 ½ tablespoon each) 2 inches apart onto the baking sheets. Press one chocolate chip, pointed-side down, into the top of each cookie. Bake, rotating the sheets halfway though, until golden brown around the edges, 12 to 14 minutes. Let cool slightly on the baking sheets, then transfer to a wire rack to cool completely.
Place the remaining 1/4 cup chocolate chips and oil in a small microwave-safe bowl. Microwave on high in 30-second intervals, stirring in between, until melted. Dip a toothpick in the melted chocolate and draw legs around the exposed chip.