Heat 2 tablespoons oil and 1 tablespoon butter in a large pan. Add onion and cook over medium-low heat for 20 to 25 minutes, stirring occasionally. Onions should be golden brown.
Add garlic and cook for 1 minute.
Stir in balsamic vinegar, sugar, and thyme and remove from heat. Season with salt and pepper.
In a medium bowl, stir together warm water, yeast and 1 teaspoon of sugar. Let sit 10 minutes.
Stir flour, 2 tablespoons of oil and 1 teaspoon of salt into the yeast mixture until a dough forms. Place dough on a floured surface and knead for 1 minute. Cover and let rest 5 minutes.
Heat oven to 425 degrees. Grease two baking sheets and divide dough in half.
Roll each ball of dough into a 14X8-inch rectangle. Bake for 10 minutes.
Top each flatbread with half of onions, pears, mozzarella cheese, and pecans. Bake for 10 to 15 minutes or until lightly browned. Sprinkle with blue cheese. Cut into wedges and serve.