Preheat the oven to 350°F. Grease a 10-inch Bundt or tube pan with the vegetable oil. Dust the bottom and sides of the pan evenly with the bread crumbs.
With an electric mixer in a medium-size mixing bowl, beat the egg whites until stiff peaks form. Set aside.
In an electric mixer fitted with a paddle attachment, cream 3/4 pound (3 sticks) of the butter and the sugar together. Add the egg yolks one at a time, beating after each addition. Add the carrots, milk, and lemon juice and beat until well mixed. Add 1 teaspoon of the salt, the baking powder, baking soda, and flour and beat again until well mixed. Fold in the beaten egg whites until no white streaks remain and pour into the prepared pan.
Bake until golden brown and the top springs back when touched, about 1 hour. Remove from the oven and let cool for about 5 minutes. With a thin knife, loosen the sides of the cake and remove it from the pan, turning it, bottom side up, onto a serving platter.
Melt the remaining 2 tablespoons butter in a large skillet over medium heat. Add the mirliton and peas, season with the remaining 1/2 teaspoon salt and the black pepper, and cook for about 5 minutes.
Spoon the mixture into the hole of the carrot ring. Slice the ring and serve with the vegetables.
Yield: about 12 servings