Heat the olive oil and butter in a wide, heavy pot over medium heat until the butter foams, about 1 minute.
Add the onions and kosher salt; cook, stirring occasionally, until wilted and translucent, 10–15 minutes.
Reduce heat to medium-low and continue cooking, stirring and scraping the pot occasionally, until the onions are deep golden brown and reduced, 30–40 minutes; adjust heat as needed to prevent scorching.
Stir in the brown sugar and black pepper until the sugar dissolves, about 1 minute. Add the balsamic vinegar, water, thyme leaves, and bay leaf; simmer uncovered, stirring occasionally, until thick, glossy, and jammy, 10–15 minutes. The mixture should hold a clean trail when a spatula is drawn through it.
Remove and discard the bay leaf. Taste and adjust salt. Cool to room temperature, then transfer to a clean jar and refrigerate. Serve chilled or at room temperature.