Preheat oven to 375°F. Bring the lamb out of the refrigerator.
Combine mint, rosemary, garlic, mustard, salt, and pepper together in a small bowl for the rub. Taste and adjust seasoning.
Evenly spread rub all over lamb rack.
Preheat a cast-iron skillet (or oven-safe sauté pan) on stovetop over medium heat. Add butter and cooking oil to hot pan. Sear all sides of lamb rack for 2-3 minutes per side.
Place pan in oven and roast for 25-30 minutes.
Once lamb is finished, remove from pan and place on a dish. While the lamb rests, create the pan sauce.
Deglaze skillet with fortified wine. Reduce by half on medium-low heat, stirring occasionally.
Add demi-glace and water. Bring to a boil then reduce heat to low and let simmer for 5-8 minutes.
Slice lamb rack and serve with pan sauce. Enjoy!