Wash the peaches, remove the pits, and cut the fruit into approximately 1/2-inch (1.25 cm) wedges. Leave the skins on for extra color and structure, or peel the peaches if you prefer a softer texture.

Place a 10- to 12-inch skillet over medium heat. Add the butter and let it melt until foamy but not browned.

Stir in the brown sugar, cinnamon, nutmeg, and salt. Cook for about 30 seconds, stirring until the sugar begins to dissolve into the melted butter.

Arrange the peach wedges in the skillet and toss gently until coated in the butter and sugar mixture.

Let the peaches cook for 3 to 4 minutes without excessive stirring. The cut surfaces should begin to soften and develop light caramelization.

Gently turn the wedges with a spatula and cook for another 3 to 4 minutes. The peaches should become tender around the edges while still holding their shape.

Continue cooking for 2 to 3 minutes, stirring gently, until the peach juices and brown sugar form a glossy syrup. If the peaches release a lot of liquid, increase the heat slightly and cook until the sauce thickens.

Reduce the heat to low: Stir in the vanilla extract and lemon juice. Cook for another 30 seconds.

The peaches should be easily pierced with a fork but should not be falling apart. The sauce should coat the fruit and the back of a spoon.

Remove the skillet from the heat and let the peaches stand for 5 minutes. The syrup will thicken slightly as it cools.

Spoon the warm peaches and their sauce into bowls or over biscuits, pancakes, pound cake, oatmeal, or ice cream.

Finish with whipped cream, toasted pecans, additional cinnamon, or vanilla ice cream if desired.