For the green goddess dressing: Combine the anchovies, garlic, yogurt, mayonnaise, parsley, basil, chives, tarragon and lemon juice in a blender and blend until smooth and green. Season with salt and pepper to taste. Refrigerate until ready to serve.
For the rice bowl: Bring a large saucepan of water to a boil. Add 1 teaspoon salt and the brown rice and cook until the rice is tender but still toothsome, about 45 minutes. Drain well.
Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the fennel, shiitakes and 1/4 teaspoon salt and cook, stirring occasionally, until the mushrooms are golden brown in spots and the fennel is crisp-tender, about 5 minutes. Add the remaining tablespoon oil to the skillet, followed by the sugar snap peas, zucchini and another 1/4 teaspoon salt. Cook until crisp-tender, stirring occasionally, about 5 minutes.
Divide the rice among 4 bowls. Top with the vegetable mixture and then with the avocado slices. Drizzle with green goddess dressing and sprinkle with the pepitas.