Cook the lobster. Remove meat from shells. Cut into large bite size pieces. Reserve in the refrigerator until needed. Cut the chicken into 1 to 2-inch pieces. Season with salt and pepper. Lightly dust with flour.
In a large skillet, heat 2 tablespoons oil. Over high heat, saute the chicken until golden in color, about 5 minutes. Remove chicken and transfer to a plate. Add the remaining 2 tablespoons oil and 2 tablespoons butter. Over medium heat, saute the potatoes, pearl onions, and carrots for 1 to 2 minutes. Add garlic, red pepper flakes, thyme, and bay leaf. Continue to saute until vegetables are glazed, about 2 to 3 minutes.
Deglaze with white wine and reduce by half. Add the tomato paste, chicken stock, and 1 cup cream. Bring to a boil, and reduce by half. Add the reserved chicken and lobster pieces. Continue to cook and stir until well incorporated. Season to taste with salt and pepper. Stir in peas. Transfer mixture into 4 soup bowls or 2-cup ramekins and allow to cool.
Preheat oven to 400 degrees F. Roll out puff pastry into 1/4-inch thickness. Measure diameter of serving bowls. Cut 4 circles of puff pastry at least 1-inch bigger than diameter of serving bowls. Chill dough until the filling is cold.
To make an egg wash, beat together the egg and 1 tablespoon cream. Brush the edge of serving bowls with egg wash. Place 1 puff pastry circle on top of one serving bowl. Press puff pastry against the edge with the egg wash, being careful not to tear or create a hole in the puff pastry. Brush the top of the puff pastry with egg wash. Repeat process with the remaining 3 bowls. Bake for 20 minutes or until golden. Serve warm.
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