Preheat oven to 350 degrees F. Line a 10x15-inch rimmed baking sheet with foil and coat lightly with non-stick cooking spray.
Spread the pretzels evenly onto the foil (it's okay if they overlap).
In a medium saucepan, combine the butter and brown sugar and bring to a boil over medium heat. Boil for 2 minutes, stirring frequently, then pour directly (and evenly) over the pretzels.
Bake the pretzels in the preheated oven for 10 minutes. Set aside to cool slightly.
In the top of a double boiler, melt the white chocolate over simmering water, stirring constantly. When it's just melted, remove from the heat and carefully spread on top of the pretzels. Sprinkle the white chocolate evenly with the rainbow sprinkles.
Refrigerate the pretzel bark in the pan for 15 to 20 minutes, until the chocolate is set. (If desired, drizzle with dark chocolate. Refrigerate for an additional 15 to 20 minutes afterward.)
Bread or cut into pieces and serve. The bark can be stored in an airtight container at room temperature for up to 1 week.