Set your broiler to high.
Wash the peaches and, using a paring knife, score an "X" into the pointed end of the peach. Place them in a stainless steel bowl.
Drizzle the olive oil over the peaches. Gently toss the peaches with the oil.
Sprinkle the salt and sugar over the peaches. Now, toss them again.
Place the seasoned peaches on a broiler pan in an even layer, with good separation between the peaches--about an inch if possible. The skins will blister and "shrink" away from the peach if they're not touching. You want this.
Sprinkle the thyme leaves across the peaches. Broil the peaches until the skins char and split, about 15 minutes, turning the peaches during cooking, trying to char evenly all around.
Remove the peaches from the broiler and let cool at room temperature for about 10 minutes.
Peel the peaches. Don't stress if there are a few stray bits of skin stuck to the peaches. You'll be processing them further.
Using a food mill (preferably), blender (acceptable), or food processor (also okay), pass the slightly warm peaches and surrounding drippings through the mill. The food mill makes you look fancy, is a conversation piece, and creates textural magic.
Reserve in a refrigerator-friendly container to use as a sauce for grilled chicken or pork, or for Chicken-Pecan Meatballs. Yes, you can freeze it.