1. Bring 1/2 water and 2 Tbsp oil to a simmer in a large skillet. Add the Brussels sprouts, season with 1/2 tsp salt and cook, covered, stirring occasionally, for 5 minutes.
2. Meanwhile, in a small bowl, whisk together the jam, vinegar, mustard, and 1/2 tsp pepper.
3. Uncover, increase the heat to medium-high and cook until the water evaporates, about 2 minutes. Drizzle the remaining 2 tbsp oil over the sprouts, add the garlic and cook, tossing occasionally, until the Brussels sprouts are golden brown and tender, 2-3 minutes.
4. Add the jam mixture and the chestnuts and cook for 1 minute.