In a small saucepan set over low heat, melt the butter, swirling the pan occasionally. Once the butter is melted, stir in the bonito flakes. Turn off the heat and set aside for 10 minutes. Set a fine-mesh sieve over a medium bowl and strain the butter, using a rubber spatula to press down on the bonito flakes to extract all of the butter from them. Discard the bonito flakes and pour the infused butter into an airtight container and refrigerate until using.