To make the tea infused butter: Put the butter in the saucepan on medium-high heat. When the butter starts to melt, add the tea leaves. Stir. When the butter is hot and has all melted, remove from the heat and allow the tea to steep for about 10 minutes.
Open the fine-mesh bag over a small bowl and pour the butter through the bag into the bowl to strain. Carefully squeeze the rest of the tea-infused butter out of the bag. Put the bowl aside to cool completely, then refrigerate the butter overnight or until it "hardens" to normal butter consistency.
Remove the butter from the oven to soften a little. Preheat the oven to 180°c. Line 2 baking trays with baking paper.
Pour the butter and sugar in a mixing bowl and beat with a hand-held mixer on medium speed until the butter and sugar have formed a creamy pale mixture. Add the flours and mix well until well combined.
Flour your work surface and rolling pin and roll out the dough until it is an even height. Cut out the shortbreads using your favourite cookie cutter and place on the trays.
Bake for 10-12 minutes or until your shortbread are just starting to turn golden on the edges. Remove from the oven and allow to cool before transferring to a wire rack to cool completely.
(Optional) Melt the white chocolate using our favourite method and drizzle a little over each shortbread.
Notes
I found unsalted butter infused better with the taste of tea. However, my Grandma always used salted butter and this recipe can do with a pinch of salt. However, it depends on how subtle a tea taste you like and which tea you have chosen.