1. Heat oven to 400˚. Cut ¼" off top of garlic head. Rub garlic with 1 tbsp. canola oil; wrap in foil and transfer to oven. Cook garlic until soft, about 1 hour; set aside to let cool. Squeeze the garlic cloves from their skin and finely chop them into a smooth paste; transfer garlic to a medium bowl along with the vinegar, capers, 1 tbsp. shallots, and anchovies. Slowly whisk in 8 tbsp. olive oil and season with salt and pepper; set vinaigrette aside.
2. Season the lamb generously all over with salt and pepper. Heat the remaining canola oil in a 10" cast-iron skillet over medium-high heat. Add the lamb and cook until meat is browned on all sides, about 12 minutes. Transfer skillet to the oven and cook lamb until an instant-read thermometer inserted into the thickest part of the meat registers 130˚, 10 to 15 minutes. Transfer the lamb to a plate, cover loosely with foil, and let rest for 30 minutes. (Lamb will continue cooking to medium rare as it rests.)
3. Increase the oven heat to 475˚. In a medium bowl, toss the potatoes with 2 tbsp. of the olive oil and the rosemary; season with salt and pepper. Transfer the potatoes to a rimmed sheet pan and cook, flipping occasionally, until golden brown and tender, about 20 minutes. Set the potatoes aside.
4. Heat the remaining olive oil in an 8" skillet over medium-high heat. Add the mushrooms and cook, flipping them once, until browned and crisp, about 5 minutes. Add the remaining shallots, butter, and thyme and cook, stirring frequently, until the flavors meld, about 1 minute more. Set mushrooms aside on a plate.
5. Put the mixed greens and the herbs into a large bowl. Whisk the vinaigrette and drizzle most of it over the greens. Toss the greens and season them with salt and pepper; transfer the greens to 4 to 6 plates and divide reserved potatoes and mushrooms between them. Thinly slice lamb against the grain and divide between the plates. Sprinkle remaining vinaigrette over salads and garnish with pistachios.