Preheat the oven to 375 degrees F (190 degrees C).
Place olive oil and butter in Dutch oven over medium heat. Stir in onion and ground lamb; brown the meat, breaking it up into small crumbles as it cooks, about 10 minutes.
Stir in flour until incorporated, then mix in salt, black pepper, rosemary, paprika, cinnamon, ketchup, and garlic; cook and stir until garlic is fragrant, 2 to 3 minutes.
Stir in water and scrape up any brown bits from the bottom of the Dutch oven. Reduce heat to medium-low and bring mixture to a simmer; cook and stir until thick, about 5-6 minutes.
Remove lamb mixture from heat and stir in peas and carrots until combined.
Spread lamb mixture into the bottom of a 9×13-inch baking dish and set aside.
Place potatoes into a large pan of salted water. Bring to a boil, reduce heat to medium, and cook until tender, about 15 minutes. Drain well and return potatoes to pan.
Mash butter, cayenne pepper, cream cheese, and Irish cheese into the potatoes. Mash until combined and potatoes are smooth. Season to taste with salt and black pepper.
Whisk together egg yolk and milk in a small bowl; stir into the mashed potato mixture.
Top the lamb mixture in the baking dish with the mashed potatoes and spread evenly to cover.
Bake in the preheated oven until the top is golden brown and sauce is bubbling up around the edges, 25 to 30 minutes.
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