Brown the butter: Melt the butter in a light-colored saucepan over medium heat. Cook, stirring often, until the milk solids turn golden brown and smell nutty, 5 to 8 minutes. Pour into a heatproof bowl with all the browned bits and cool until just warm, 15 to 20 minutes.
Prep the oven and skillet: Preheat the oven to 350°F (175°C). Grease a 10- to 12-inch oven-safe skillet.
Mix the dry ingredients: Whisk together the flour, baking soda, salt, and cinnamon. Stir in the oats and set aside.
Add the sugars: Whisk the brown sugar and granulated sugar into the cooled brown butter until slightly thickened.
Add the eggs and vanilla: Whisk in the eggs one at a time, then the vanilla, until glossy and smooth.
Combine wet and dry: Fold the dry ingredients into the butter mixture with a spatula just until no dry flour remains.
Fold in the mix-ins: Fold in the chocolate chips, coconut, and walnuts. If the dough is very stiff, mix in 1 to 2 teaspoons of milk.
Press into the skillet: Press the dough into an even layer in the skillet. Press a few extra chocolate chips on top.
Bake: Bake for 22 to 30 minutes, until the edges are deeply golden and the center is puffed but still a little soft. For a gooier center, start checking at 20 minutes.
Finish and serve: Sprinkle with flaky sea salt, if using. Cool in the skillet for 10 to 15 minutes, then serve warm with vanilla ice cream.