Preheat oven to 350 degrees. Butter and flour two 8-inch-round-by-2-inch-high cake pans; set aside.
In a small bowl, whisk lemon-thyme with flour, salt, and lemon zest; set aside.
Using an electric mixer on high speed, beat butter and sugar until light and fluffy. Add eggs one at a time, beating well after each addition; add vanilla and lemon zest. With mixer on low, gradually add the flour-lemon-thyme mixture, beating just until combined (do not overmix).
Divide batter between cake pans and smooth tops with a spatula. Bake until a toothpick inserted in the centers comes out clean, 30 to 40 minutes. Let cool in pan 15 minutes, then invert onto a wire rack, and turn upright to cool completely.