I have a question about the recipe "Pear-Filled Cowboy Coffee Cake" from AntoniaJames. alternate tools
I would whisk the dry ingredients by hand, un-clumping the brown sugar with your fingertips if necessary. Then use handheld beaters to mix in the butter, which should be completely at room temperature. You will want to be very careful not to overmix the batter though -- just until combined and no more, or the cake will be tough.
It seems to me that the butter is to be cut into the dry ingredients as is done for pie crust or biscuit dough. You could use a pastry blender as pictured, or a strong fork or just work it in with your finger tips until it is a sandy consistency before adding the liquid (and not over-mixing). For that technique to work well, the butter should be cold, and cut into small cubes, so that it doesn't mush into the dry ingredients.
Another option would be to whisk the flour mixture, then grate in very cold butter. A small amount of mixing afterward should get the butter incorporated.