Preheat the oven to 350°F (175°C). Place a 9-inch square baking dish on a rimmed sheet pan.
Heat the olive oil in a large saucepan over medium-high heat. Add the shallot, fennel, celery, and 1/2 teaspoon salt and cook, stirring often, until the fennel softens and the shallots turn golden-brown, about 8 minutes. Add the ground lamb, dried herbs, and garlic powder and cook, stirring continuously to break apart the meat while it browns, about 8 minutes more.
Stir in the tomato paste until combined. Add the beef stock and red wine vinegar. Bring to a boil then reduce the heat to medium-low and simmer, uncovered, until the liquid has reduced, about 20 minutes. Stir in the green beans, 1 teaspoon salt, and 1/2 teaspoon pepper, then let the mixture simmer for 5 minutes, stirring occasionally. Remove the pan from the heat and spoon the mixture into the casserole dish. Set aside.
In a large pot with a lid, combine the cauliflower florets and 1 inch of water, then bring the water to a boil over high heat. Cover, let the cauliflower steam until very tender when pierced with a paring knife, 8 to 10 minutes. Carefully pour off the water to drain the cauliflower. To the pot of cauliflower, add the butter, 1 teaspoon salt, and 1/8 teaspoon pepper. Use an immersion blender or food processor to blend the cauliflower into a smooth puree. Add the egg and the egg yolks and puree until well blended and smooth.
Spoon the cauliflower puree over the lamb mixture. Sprinkle with paprika, transfer the baking pan with the casserole to the oven, and bake until the mash is golden brown and the filling is bubbling around the edges, 30 to 35 minutes. Serve.