Cut unsalted butter into chunks and brown over medium heat in a small saucepan, stirring frequently, until deep amber with a nutty aroma, about 5 minutes.
Preheat oven to 350°F and line a baking sheet with parchment paper.
Pour browned butter with browned bits into a bowl and let cool about 15 minutes.
In a separate bowl, whisk together flour, baking soda, and salt; set aside.
Roughly chop the Cadbury mini eggs and combine with chocolate chips in a small bowl.
Stir brown sugar into the cooled brown butter until combined and slightly lighter, about 1-2 minutes.
Add the room temperature egg and vanilla extract to the butter mixture; stir until smooth.
Gradually add dry ingredients to wet mixture, stirring just until combined to avoid overmixing.
Fold in chopped Cadbury eggs and chocolate chips gently to preserve candy pieces.
Drop 3-tablespoon portions of dough onto prepared baking sheet, spacing 2 inches apart.
Bake for 10-12 minutes until edges are set but centers remain slightly underbaked.
Cool cookies on baking sheet for 10 minutes before transferring to a cooling rack to cool completely.