In a medium sauté pan over low heat, add olive oil, onions, and thyme. Caramelize slowly, stirring occasionally for 15-20 minutes or until onions are tender and golden brown. Remove from heat and let cool.
Heat oven to 350ºF. Place flatbread bases on a baking sheet and toast in oven for 5 minutes, or until lightly crispy.
Spread 2 tablespoons of yogurt onto each toasted flatbread base. Top evenly with caramelized onions, pears, figs, and blue cheese.
Bake flatbreads for 6-8 minutes or until crust is crispy and cheese is melted. Remove from oven and let cool slightly.