Preheat and Prepare: Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper to prevent sticking.
Mix Dry Ingredients: In a medium bowl, whisk together flour, baking powder, baking soda, and salt until evenly combined. Set aside.
Cream Butter and Sugar: In a large bowl, cream softened butter and granulated sugar together for 2-3 minutes until light and fluffy.
Add Wet Ingredients: Add egg and vanilla extract to the butter mixture. Beat until well combined and smooth.
Combine Mixtures: Gradually fold the dry ingredient mixture into the wet ingredients. Mix just until combined, avoiding overmixing.
Portion and Bake: Using a cookie scoop or tablespoon, portion dough onto prepared baking sheet, leaving 2 inches between each cookie. Bake for 10-12 minutes until edges are lightly golden.
Create Indentations: Immediately after removing from oven, press a small jar or spoon into the center of each hot cookie to create a well. Let cool on baking sheet for 10 minutes.
Cool Completely: Transfer cookies to a wire rack and let cool completely before filling, about 30 minutes.
Fill with Custard: Once cookies are completely cool, fill each indentation with vanilla custard using a piping bag or small spoon. Be generous but avoid overflow.
Prepare Chocolate Drizzle: In a microwave-safe bowl, combine chocolate chips and vegetable oil. Microwave in 15-second intervals, stirring between each, until completely melted and smooth.
Drizzle and Set: Drizzle melted chocolate over filled cookies in desired pattern. Let chocolate set for about 15 minutes before serving.