Heat a large cast iron skillet over medium heat. Add the scallions and char, tossing occasionally until charred all over, about 4 minutes.
Make the butter
Add the butter, scallions and salt to the bowl of a food processor. Pulse until combined. Add to a small bowl until ready to serve.
Mix the dry ingredients
Heat oven to 400ºF (200ºC). In a large mixing bowl, whisk together the cornmeal, flour, sugar, salt, baking powder, and baking soda.
Mix the wet ingredients
In another medium mixing bowl, whisk together the melted butter, buttermilk and eggs.
Fold the wet ingredients into the dry until just combined. Gently stir in the cheese.
Add batter to muffin tins
Butter a large muffin tin then spoon ½ cup of the batter into each cup of the prepared muffin tin.
Bake until the muffin tops are golden brown and cooked through so a toothpick inserted into the center comes out clean, about 20 minutes.
Let cool on a wire rack, then carefully remove from the muffin tins. Serve at room temperature or warm with scallion butter.