1. Cut shallot lobes in half lengthwise, then cut each half into thick wedges.
2. Heat a large (not nonstick) frying pan over medium heat, then swirl in 2 tbsp. oil and the butter. When butter begins to foam, add shallots and stir to coat. Season with 1/2 tsp. salt and a pinch of pepper.
3. Reduce heat to medium-low and cook shallots 15 minutes, stirring often. Reduce the heat to low and cook, stirring every 2 to 3 minutes, until completely caramelized (deep golden brown), 15 to 25 minutes. Transfer to a plate to cool.
4. Meanwhile, roll cheese neatly in plastic wrap to create a uniform log shape. Twist ends of the plastic wrap tight and roll log back and forth on a work surface a few times to smooth out surface of cheese and give it an even shape.
5. Cut figs into halves or quarters, depending on their size. Lay cut side up and season with 1/2 tsp. salt, pepper, and about 1 tbsp. oil.
6. Cut radicchio lengthwise into 1/3-in.-thick ribbons and put in a large bowl.
7. Toss radicchio with caramelized shallots, parsley, and salt and pepper to taste. Drizzle lemon juice, remaining 1 tbsp. oil, and 2 1/2 tsp. saba over salad. Toss to combine.
8. Run a thin, sharp knife under hot water for a few seconds, then wipe it dry. (A hot knife will slice soft goat cheese more cleanly.) Remove plastic wrap from cheese and slice cheese into 12 rounds (about 1/3 in. thick), slicing straight down and reheating knife between slices. Season with pepper.
9. Heap radicchio salad on a long platter and tuck figs in between leaves. Place goat cheese on salad, slightly overlapping the slices. Drizzle a bit of saba and oil over goat cheese and serve.
*Saba, also known as mosto cotto, is the syrupy, cooked-down juice of freshly pressed red grapes. It's been made in Italy since Roman times. Find it in well-stocked grocery stores and online.