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