Preheat the oven to 400°F and grease a baking dish or cast iron skillet. Place baking dish or skillet onto a larger tray to catch any spillage in the oven.

Melt the butter in a deep pan over medium heat. Toss in the carrots, onion, and celery and let them soften, about 5 to 7 minutes.

Add the garlic, sage, thyme, and rosemary and cook until the whole pan smells fragrant, about 1 minute.

Dust the flour over the vegetables and stir it in, letting it cook for a minute or two so it loses that raw flour taste.

Whisk in the chicken broth a little at a time, followed by the half and half. Season with the salt, black pepper, smoked paprika, red pepper flakes, yellow mustard, turmeric, and nutmeg.

Bring it to a simmer and let it thicken up, about 5 minutes.

Stir in the shredded chicken and peas and take the pan off the heat.

Start out by browning the butter. To a pan over medium heat, add the butter. Cook for 5 to 6 minutes until it starts to bubble and then brown. Make sure not to cook it any further or it can burn. Transfer it to a bowl and allow it to cool so it's not hot.

In a bowl, whisk together the flour, corn meal, salt, baking powder, baking soda, white sugar, and brown sugar.

To the same bowl, add the cooled browned butter, egg, and buttermilk. Whisk again just until combined, no more.

Fold in the cheddar and diced jalapeño.

Spoon the cornbread batter over the chicken filling in the skillet and spread it out evenly to the edges.

Bake for 20 to 25 minutes, until the top is golden and a toothpick pulls out from the center clean.

Give it 5 to 10 minutes to rest before digging in.