Peel potatoes if you have to (I don’t!), chop into 2″ chunks. Boil in a large pot of water until tender.
While potatoes are boiling, heat olive oil in a large pan over medium high heat. Add onions, cook until tender and translucent.
Add ground beef, brown all over, stirring and breaking up meat chunks frequently.
Add the grated carrot, parsnip, and turnip, as well as the garlic, stirring well to combine.
Once meat is cooked through and veggies are tender, whisk together broth and corn starch until smooth.
Pour over meat and veg mixture, season with rosemary, pepper, savory, salt, and parsley. Stir until well combined and broth mixture starts to thicken. Remove from heat.
Once potatoes are tender, drain and transfer to a large bowl – I like to use my stand mixer. Mash/whip for 30 seconds or so, until potatoes have broken down a bit. .
Add butter, and broth, continue to beat until smooth. Season with salt and pepper
Preheat broiler to high, spray a 9 x 13" baking pan with nonstick spray
Transfer meat mixture to pan, spreading to form an even layer.
Spread potato mixture evenly over the meat mixture, broil until potatoes are as browned as you would like them.
Serve immediately, or cool to room temperature, cover with plastic wrap, and chill