Heat broth in a saucepan over medium heat; add rice and simmer until rice is slightly tender, 10 to 12 minutes.
Heat butter and oil in a skillet over medium heat; cook onions in the hot butter-oil mixture, stirring constantly, until onions are golden brown, 15 to 20 minutes. Reserve about 1 tablespoon caramelized onions for garnish; spoon the remaining caramelized onions into the simmering broth.
Mix yogurt, lemon juice, and salt in a bowl; whisk flour into yogurt mixture until smooth. Blend yogurt mixture and 1 cup broth (free of onion and rice) in a blender until smooth.
Pour blended yogurt mixture into simmering broth; bring to a boil. Reduce heat and simmer until flavors have blended, 2 to 3 minutes.
Garnish with reserved caramelized onion and toasted pine nuts.