How to Make It
In a medium saucepan, heat the olive oil. Add the shallot and garlic and cook over moderately low heat, stirring frequently, until softened, about 3 minutes. Add the lentils and stock and bring to a boil. Simmer over moderate heat, stirring occasionally, until all of the stock has been absorbed and the lentils are tender, 20 to 25 minutes. Stir in the miso and soy sauce and season with salt. Keep warm.
In a small saucepan, combine all of the ingredients with 1/4 cup of water and bring to a simmer. Cook over moderate heat, stirring occasionally, until the sauce reduces to a glaze, 15 to 20 minutes. Remove from the heat.
In a small saucepan, heat the olive oil. Add the shallot and garlic and cook over moderately low heat, stirring frequently, until softened, about 3 minutes. Add the stock and cream and bring to a simmer. Cook over moderate heat, stirring occasionally, until the cream is thickened to a sauce consistency, about 20 minutes. Transfer the mixture to a blender and add the lemon juice. Puree until smooth. Season the garlic cream with salt and pepper and scrape it into a small bowl; keep warm.
Preheat the oven to 350°. In a large skillet, melt the butter in the olive oil. Add the mushroom caps and thyme and cook over moderate heat until golden, about 3 minutes per side. Season the mushrooms with the lemon juice, salt and pepper. Arrange the mushrooms gill side down on a baking sheet and brush with some of the teriyaki glaze; reserve any remaining glaze for another use. Roast the mushrooms until just tender, about 5 minutes. Let rest for 2 minutes.
Slice the mushrooms 1/4-inch thick, keeping them intact. Using the palm of your hand, press and turn the mushrooms slightly to fan them out on plates. Scatter the lentils on top and dust with the smoked paprika. Spoon the garlic cream around the mushrooms and serve.