Char the peppers on a grill, under a broiler, or directly over a gas flame, turning every 3 to 5 minutes, until the skins are completely blackened, about 10 to 15 minutes total. Once charred, place the peppers in a medium bowl and tightly cover with foil until cool enough to handle.
Whisk the flour, cornmeal, baking powder, sugar, salt, and chile flakes in a large bowl. Cut in butter until pea-size pieces form and the mixture resembles breadcrumbs.
Once cool enough to handle, remove the stems, skins, and seeds from the peppers. Dice the flesh and whisk into the buttermilk in a small bowl.
Toss cheese with the remaining teaspoon of flour to prevent sticking. Fold into the flour mixture until evenly distributed, then add the buttermilk and peppers, mixing with a rubber spatula until the dough comes together and no dry spots remain.
Using a 1⁄3-cup measuring scoop, arrange 10 biscuits onto two parchment-lined baking trays, spacing 2 inches apart. If the dough is sticking to the measuring cup, use a rubber spatula to loosen.
Bake at 425°F for 15 to 18 minutes, rotating halfway through, until golden brown and a toothpick inserted into the middle comes out clean. Cool for 15 minutes before serving.