Preheat your oven to 400°F (200°C).

In a small bowl, combine the mashed black garlic, softened butter, parsley, lemon juice, and lemon zest. Mix until well blended. Set aside to allow the flavors to meld.

Season the rack of lamb generously with salt and pepper.

In a large skillet over medium-high heat, heat 2 tablespoons of olive oil. Add the minced garlic and sauté until fragrant, about 1 minute.

Add the tarragon and breadcrumbs to the skillet; mix well and cook for another 2-3 minutes until the breadcrumbs are golden and toasted. Remove from heat.

Brush the rack of lamb with Dijon mustard, then press the breadcrumb mixture onto the meat, ensuring an even coating.

In the same skillet, add the remaining 2 tablespoons of olive oil and heat over high heat. Sear the lamb rack on all sides until nicely browned, about 2-3 minutes per side.

Transfer the seared lamb rack to a baking dish and roast in the preheated oven for 15-20 minutes for medium-rare, or until the internal temperature reaches 130°F (54°C).

Remove the lamb from the oven and let it rest for 5-10 minutes before slicing.

Serve the lamb rack slices with dollops of the black garlic compound butter on top.