Heat the milk in a large Dutch oven over medium heat, stirring frequently so that the bottom doesn’t scald, until it reaches 160°F, 20 to 25 minutes.

Whisk in the yogurt and continue cooking, stirring frequently, until the milk mixture reaches 190°F, 8 to 10 minutes more. Remove from the heat and whisk in the vinegar. Let the milk mixture sit undisturbed to let the proteins coagulate into curds, separate from the yellow watery whey and rise to the surface, 5 to 7 minutes.

Meanwhile, line a fine-mesh sieve with the cheesecloth and set it over a medium bowl. Gently skim the solid curds from the surface with a slotted spoon and transfer them to the sieve. Discard the whey or retain it for another purpose. Let the curds drain until cool enough to handle, about 20 minutes.

Season the cheese with salt to taste and serve warm. Alternately, transfer the cheesecloth with the cheese to a small wicker basket or plastic cheese mold. Fold the edges of the cloth over the cheese and refrigerate until set, at least 2 hours or overnight. Unwrap and invert the cheese onto a plate, removing the cheesecloth and mold before serving.