The day before you want to serve this dish, make the vanilla-infused olive oil: scrape the two halves of the vanilla pod with the back of a paring knife and place the beans (a paste, really) into the olive oil. Cover tightly and reserve.
The next day, stir the infused oil, dredging the vanilla beans settled at the bottom. Heat 3 tablespoons of the vanilla oil in a medium non-stick pan over medium-high heat until the oil shimmers. Add the fig slices and cook for 2-3 minutes, tossing and turning very gently. The more ripe figs will break apart, but some slices should remain whole. Turn the figs onto a platter, and let cool.
Divide the arugula evenly between 4 small plates, and drizzle with the remaining olive oil (about 1 teaspoon per serving). Spread the goat cheese onto the toast rounds and set aside. Divide the cooked figs between the four plates, placing them on the arugula.
Place the chèvre toast round along one side of the plate, and drizzle the figs with any remaining olive oil from the pan. Serve immediately. Bon app’!