In a medium saucepan over low heat, melt the butter.
Remove from heat and scatter 1 c. white chocolate chips over the butter. Do not stir.
Allow to sit for 2 minutes, then gently stir the chips and butter together until mostly combined. Set aside to cool to lukewarm, about 10 minutes.
Meanwhile, in a large bowl, whisk together flour, brown sugar, baking powder, and salt.
Add the eggs, vanilla, and white chocolate mixture to the flour mixture. Mix until just combined.
Fold in the remaining 1 c. white chocolate chips.
Fold in the Lucky Charms marshmallows.
Spread into a greased 9x13-inch pan.
Bake at 325 degrees for 20-22 minutes, or until edges are golden brown and a toothpick inserted near the center comes out clean.