Heat the oven to 425°F. In a large bowl, whisk together the flour, salt, and pepper. In a separate bowl, whisk together the eggs and milk. Whisk the eggs into the flour mixture until just combined. Then stir in the thyme and chives.
Melt the butter in a 12-inch cast-iron or other oven-safe skillet over medium-high heat. Continue to cook until the butter smells nutty and turns brown, 5 to 7 minutes; then swirl the skillet so the butter coats the bottom of the pan.
Pour the batter into the skillet, and scatter the cheese and flaky sea salt over the top. Bake until the Dutch baby is puffed and golden, 20 to 25 minutes. Baking it a little less gives a softer interior though less rise; baking it a little more gives you more puff and a drier interior; both ways are good.
Serve immediately, garnished with thyme and chives, and with Sriracha and/or lemon wedges on the side if desired.
