Halve the peaches and remove the pits with a paring knife. Set the halves cut side up on a cutting board.
Melt the butter in a 12-inch / 30 cm skillet over medium heat until fully melted, about 1 minute.
Stir in the brown sugar, cinnamon, and salt. Cook for 1 to 2 minutes, stirring, until the mixture is smooth and bubbling.
Lay the peach halves cut side down in the skillet in a single layer, spacing them so they don't touch.
Cook undisturbed for 3 to 4 minutes until the cut sides are deep amber and pull away from the pan easily.
Flip each peach half gently with a spoon or tongs. Cook 2 to 3 minutes more until the peaches are tender but still holding their shape.
Remove the skillet from the heat. Stir the vanilla extract and lemon juice into the caramel left in the pan.
Spoon the warm caramel over the peaches and finish with a pinch of flaky sea salt if using.
Serve warm, straight from the skillet or plated with your topping of choice.