Preheat the oven to 350 degrees.
If using an 8x8-inch pan, grease with oil or butter. If making corn muffins, generously grease muffin tin with oil or butter, or line the pan with cupcake liners.
In a large bowl, whisk together the flour, corn meal, sugar, salt, and baking powder to break up any lumps.
In a separate small bowl, combine the melted butter, milk, eggs, and vegetable oil.
Pour the wet ingredients into the dry ingredients and stir just until combined. Do not overmix.
Pour the batter in the prepared pan or muffin tin (2/3 full) and bake until the edges are slightly golden and a toothpick inserted in the center comes out clean (or with a few moist crumbs). For an 8x8-inch pan, that will be 35-40 minutes; for corn muffins that will be 15-20 minutes.
Cool for a few minutes before slicing and serving. It's best enjoyed warm, with a drizzle of honey or a pat of butter melting into each slice.