Start ricotta: Pour milk, half-and-half, and salt into a large heavy-bottomed pot and bring to a bare simmer over medium heat, stirring every now and then to prevent scorching, 25 to 30 minutes.
Meanwhile, cook figs: Melt butter in a medium nonstick frying pan over medium heat. Slice figs in half and set cut side down in butter in a single layer. Pour in 2 tbsp. water and sprinkle with sugar. Let figs cook until they start to soften, about 3 minutes. Reduce heat to low.
Pour in mosto cotto. Let figs simmer until quite soft and mosto cotto is as thick as honey, about 7 minutes. Add balsamic vinegar and remove from heat.
When milk reaches a bare simmer, pour in enough lemon juice to separate it into thick, fluffy white curds and watery, greenish whey (if this doesn't happen immediately, add more lemon juice). Push curds gently once or twice but resist the urge to stir--that will make them gritty. Turn off heat.
Scoop ricotta into a medium-mesh colander set in the sink. Let liquid drain 10 to 15 minutes; ricotta should be moist but not wet.
Spoon about 1/3 cup ricotta onto each of 4 small plates and add 2 or 3 fig halves. Drizzle with syrup from pan and serve.
*Look for reasonably priced cask-aged balsamic vinegar, sometimes labeled “condimento.” Avoid cheap “instant” balsamic vinegar, which has colorants and sugars.
Make ahead: Ricotta and figs, each up to 2 hours at room temperature, covered, or up to 1 day, chilled (let them come to room temperature before serving).