To prepare the balsamic syrup, in a small saucepan, stir together the brown sugar, balsamic vinegar, red pepper flakes, and thyme. Bring to a simmer over medium heat and simmer until the mixture has reduced by one-third and is syrupy enough to coat the back of the spoon, about 20 minutes.
Let cool to room temperature and then strain. Store the syrup in a covered jar at room temperature for up to 1 month.
Bring a medium saucepan of water to a simmer.
Place the onions in a medium, heat-proof bowl and cover them with the simmering water.
Let sit for 10 minutes and then slip off their skins, using a paring knife to loosen the ends if necessary.
In a deep, medium saute pan, melt 4 tablespoons of the butter over medium heat.
Add the onions and cook, stirring often, until they are golden brown on both sides, 5 to 10 minutes.
Add the vinegar and scrape up the browned glaze from the bottom of the pan. Bring to a simmer over medium heat and add the remaining 4 tablespoons of butter. Reduce the heat to low, cover the pan tightly, and simmer until the onions are tender when pierced with the tip of a knife, 20 to 30 minutes.
Remove the pan from the heat and let them sit, still covered.
Generously brush both sides of the bread with oil. Toast the slices in a large skillet over medium heat until they are crispy and golden on both sides, 4 to 5 minutes per side. Divide them among 6 serving plates.
In a large bowl, toss the arugula with enough of the warm onion cooking liquid to make the leaves glossy. Divide the arugula among the pieces of toast. Top each with 3 onions and drizzle with about 1 tablespoon of the balsamic syrup.
