Adjust the oven rack to the upper middle position and preheat the oven to 375° F.

Filling:

Heat the butter in a large skillet over medium-high heat until foaming.  Add the carrots and onions and cook, stirring occasionally, until soft, about 10 minutes.  Add the meat, ½ teaspoon salt, and ½ teaspoon pepper and cook, breaking up the meat into small pieces, until browned, about 15 minutes.  Add the flour and the tomato paste and stir for about 1 minute, or until the tomato paste begins to darken.

Add the cream and cook until it spatters, about 1 minute.  Add broth, beer, soy sauce, and thyme and simmer over medium heat, stirring frequently, until the mixture is thick, about 15-20 minutes.  It should still be saucy.  Remove from heat, stir in peas, and pour into a broiler-safe, 2-quart casserole dish.

Topping:

Bring the potatoes, ½ teaspoon salt, and enough water to cover by one inch to a boil in a large saucepan over high heat.  Reduce heat to medium-low and simmer until potatoes are tender, about 20 minutes.  Drain the potatoes, return them to the saucepan and mash them with the butter and cream until smooth.  Add salt and pepper to taste.

Assembling:

Spread the potatoes over the filling using a rubber spatula.  You want to completely cover the filling to prevent it from boiling over in the oven, so start at the edges of the casserole, then fill in the center.  Smooth the filling, brush with the beaten egg, and drag a fork through the topping to make ridges.  Bake until the filling is bubbling, about 15 minutes.  Turn on the broiler and broil until the top is golden brown, about 5 minutes.  Let cool for 10 minutes and then serve.