Prepare a bain-marie, or water bath, by half filling a roasting tin with water.
Melt sugar in a nonstick pan over medium-low heat, stirring constantly until melted, 7 to 10 minutes; be careful to keep it from burning. Pour melted sugar into a flan or pie dish, coating the sides to ensure that the egg/milk mixture in the next step will not touch the container.
Pour sweetened condensed milk into a bowl. Fill the empty tin with milk and add to the bowl; stir in eggs and vanilla extract. Blend well. Fold mixture with a spatula or tap against the counter to remove air bubbles.
Pour milk mixture into the slightly cooled flan mould. Cover the flan dish with a lid or plate and place inside the water bath; don't let the water go over the rim.
Bake the quesillo in the bain-marie in the preheated oven for 45 minutes. Remove lid carefully; continue baking until set, about 15 minutes more.
Let quesillo cool to room temperature, at least 25 minutes; refrigerate 8 hours to overnight. Slide a knife around the edges of the quesillo to loosen and invert onto a plate.