Heat grill to high. Brush both sides of mushrooms with oil and season with salt and pepper. Grill caps until golden brown on both sides and just cooked through, 10 to 12 minutes. Fill the caps with the wild rice pilaf, drizzle the piquillo vinaigrette and sprinkle tops with the toasted almonds.
Place wild rice in a colander and rinse well with cold water. Combine 3 cups of the stock and the water in a large saucepan with a tablespoon of salt and bring to a boil. Add the rice and cook until the grains open and are very soft, 1 1/2 to 2 hours. Drain well.
Heat oil in a large saute pan over high heat. Add the onion and cook until soft. Add the garlic and cook for 1 minute. Add the wine and cook until reduced by half. Add the cooked wild rice, thyme, rosemary and remaining 1/2 cup of stock and season with salt and pepper and cook until heated through and flavors combine, 15 minutes. Remove from the heat and fold in the parsley.
Combine peppers, onion, garlic, vinegar, honey, mustard, and salt and pepper in a blender and blend until smooth. With the motor running, slowly add the oil and blend until emulsified.