Line a sieve with cheesecloth and rest it over a bowl that's slightly smaller than the sieve, so that there's space between the bottom of the sieve and the bottom of the bowl. Add the yoghurt, and allow to strain in the fridge for 4 hours up to overnight.
In a medium-sized pan over medium heat, stir together the honey, water, and olive oil. Add the thyme sprigs, bring to a gentle simmer, and cook for 3 minutes. Add the butter and cook another 2 minutes, until the butter is melted completely into the sauce. Remove the thyme sprigs and discard.
Halve the figs, and add them to the honey thyme sauce cut-side-down. Cook for 2-3 minutes, until softened. Gently flip (I find chopsticks helpful for this) and cook for another half minute.
Divide the yoghurt into 4-6 bowls, and top with the figs. Spoon the warm sauce overtop, and be generous with it! Best eaten right away, but swirled-together leftovers with some granola make a lovely breakfast too.