1. To prepare chutney, heat oil in a large nonstick skillet over medium-high heat. Add carrot, shallots, garlic, and ginger to pan. Reduce heat to low, and cook 10 minutes, stirring occasionally. Add sugar, honey, butter, cardamom, and thyme; cook 1 minute, stirring constantly. Stir in broth; bring to a boil. Reduce heat, and simmer 45 minutes or until carrot is tender and liquid almost evaporates. Discard thyme sprigs and ginger. Stir in 1/2 teaspoon salt; cool.
2. To prepare injera, weigh or lightly spoon flours into dry measuring cups; level with a knife. Combine flours, baking soda, and 1 teaspoon salt in a large bowl; stir with a whisk. Combine club soda and yogurt in a small bowl, stirring with a whisk until smooth. Add the yogurt mixture to the flour mixture; stir with a whisk until smooth.
3. Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Pour about 1/3 cup batter per flatbread onto pan in a spiral, starting at the center; cook 20 seconds. Cover the pan; cook an additional 40 seconds or just until set. Transfer to a plate, and cover with a cloth to keep warm. Repeat procedure with remaining cooking spray and batter, wiping the pan dry with a paper towel between flatbreads. Serve flatbreads with chutney.