Note about preheating the oven: About the time you put your potatoes in the boiling water, preheat the oven to 350 degrees. Grease a 9” baking dish; set aside.
Begin to boil salted water for the potatoes as you prep your veggies (peeling, dicing, etc.). Placing a lid on your pot will bring the water up to boiling quicker.
Heat the butter and oil in a large skillet over medium heat. Once the butter is melted and beginning to bubble, add the onions and carrots. Allow them to slightly caramelize, stirring occasionally approximately 7 to 10 minutes.
Add the garlic and stir for about 30 seconds. Scrape the veggies onto a plate and set aside.
Add the ground beef and cook until no pink remains. Drain well.
While the beef is draining, add the tomato paste, rosemary, thyme, and paprika to the skillet. Stir continuously until the tomato paste begins to smell very fragrant and has darkened slightly in color.
Add the beef and sautéed vegetables back into the skillet and combine all the ingredients very well.
Sprinkle the flour over the beef mixture and stir for 2 minutes. You should not see any flour in your mixture and it should look dry.
Add in the beef stock, worcestershire sauce, and peas. Stir until all is combined. Raise the heat to medium-high and stir often until the mixture simmers and thickens.
Taste for seasoning and add salt and pepper as desired.
Add the potatoes to the boiling water and cook until they are easily pierced with a fork. Drain them (or use a large slotted spoon/spider to remove them), discard the cooking water, and place the potatoes back into the pot. Add the butter, salt and pepper to taste. Mash the potatoes with a potato/meat masher until creamy, or until the desired consistency.
Pour the beef-vegetable mixture into the greased baking dish, smoothing the top.
Dollop the potatoes on top and smooth them across the top. (At this point you can create designs with a spoon or fork. It’s not necessary, but creates a pretty presentation.)
Evenly sprinkle the Parmesan on top and bake for 25 minutes, or until the mixture is bubbly and hot. You can broil the Shepherd’s Pie if you’d like the potatoes a little browned, but keep watch that you do not burn the top.
Remove from the oven and serve warm.