In a 4- to 5-quart pan over medium-high heat, stir pine nuts in 1 teaspoon oil until lightly toasted, about 4 minutes. Pour into a bowl.
Pour 1 tablespoon oil into pan and add quinoa. Stir over medium-high heat until quinoa smells toasted, about 3 minutes. Add tomatoes and broth; bring to a boil over high heat. Cover, reduce heat, and simmer, stirring occasionally, until quinoa is tender, about 10 minutes. Remove from heat and stir in pine nuts, green onions, basil, and cilantro; season with vinegar and salt. Cover and let stand.
Drain chilies and split open lengthwise; discard seeds. Arrange half the chilies in a 2-quart casserole (8 by 12 in.). Scrape quinoa mixture over chilies; spread level. Cover with remaining chilies; sprinkle with cheese.
Bake in a 375° oven until casserole is hot in the center and cheese is melted, about 20 minutes.
Cut into portions and scoop out with a spoon. Add salt to taste.