Thoroughly grease a straight-sided 9-inch square cake pan with butter.
Melt the butter in a large, heavy-bottomed pot set over medium heat. Cook, swirling the butter around in the pot, until it foams and turns a medium golden brown, 3 to 4 minutes. Add the regular marshmallows and stir until they're completely melted. Add the salt and give it another stir.
Remove the pot from the heat and pour in the cereal. Gently fold it into the marshmallow until well combined. Next, add the mini rainbow marshmallows and fold them into the mix. Some will melt, but some will remain whole; that is the way you want it.
Turn the mixture into the buttered cake pan and press flat using a wooden spoon or spatula. Sprinkle with the rainbow sprinkles over the top while it is still warm. Let cool completely, about 20 minutes.
When the crispy treats are cool, place the white chocolate in a heatproof bowl set over boiling water. Heat, stirring, until the chocolate is melted and completely smooth. Set aside for a minute to cool.
Drizzle the white chocolate over the cooled crispy treats and let the chocolate set for 5 minutes. Cut into 1 1/2-inch squares.