Preheat the oven to 375 (F), and line a muffin tin with muffin cases.
Using a handheld electric whisk or a freestanding electric mixer with a paddle attachment, beat 6 tablespoons butter, 1 1/3 cup sugar, 1 1/2 cups flour, 1/4 teaspoon salt, and 1 tablespoon baking powder on low speed until the ingredients are well incorporated and have the texture of fine breadcrumbs.
Break 2 eggs into a Pyrex measuring cup, then add 1 cup whole milk and 1/2 teaspoon vanilla and whisk together by hand.
With the mixer running on a low speed, slowly pour three-quarters of the wet ingredients into the flour and butter mixture. Once everything has been mixed together, scrape down the sides of the bowl to pick up an ingredients that were missed, then add the rest of the milk mixture and continue mixing the batter until it is smooth and even.
Divide the batter between the paper cases, filling each case with 2 tablespoons of batter (or, filling each up to two-thirds full).
Bake for 18-20 minutes in the preheated oven until well risen and the tops bounce back when touched. Leave to cool slightly before removing them from the tin, then place on a wire rack to cool completely while you make the coffee soaking syrup.