Heat butter and 1 tablespoon olive oil in a pot over medium heat. Add shallots, and cook, stirring frequently, until golden, 12 to 14 minutes. Transfer shallots to a bowl. Add wild rice to pot, and stir for 1 minute. Add stock and 1/2 teaspoon salt. Bring to a boil, and stir again. Cover, reduce heat, and gently simmer until grains burst, about 40 minutes. Uncover, and cook until liquid has been almost completely absorbed, about 15 minutes more. Spread on a baking sheet to prevent rice from overcooking.
Meanwhile, in a separate pot, heat 1 tablespoon olive oil over medium heat. Add basmati rice, and stir for 1 minute. Stir in the water and 1/2 teaspoon salt, and bring to a boil. Cover, reduce heat, and gently simmer for 10 minutes (do not uncover). Remove from heat, and let stand, covered, for 3 minutes. Spread on a baking sheet to prevent rice from overcooking.
Whisk vinegar, sesame oil, remaining 3 tablespoons olive oil, and salt and pepper to taste in a large bowl. Stirring constantly, gradually add wild and basmati rice. Add shallots, pomegranate seeds, pistachios, and herbs. Season with salt and pepper. Serve immediately.