Preheat oven to 350°F (175°C). Grease a 9-inch round pan and line the bottom.
Whisk flour, baking powder, and salt. Cream butter and sugar until light. Beat in eggs one at a time, then vanilla.
Alternate adding dry mix and milk, starting and ending with dry. Pour batter into pan and bake 25–30 minutes until a toothpick is clean. Cool completely.
Whip cream cheese, heavy cream, and powdered sugar until spreadable. Spread a thin layer over the cooled cake so some cake shows through.
Drag the back of a spoon around the sides to create a driftwood texture. Top with thyme sprigs arranged casually.