Put 1 tablespoon of the olive oil in a wide-bottomed pan and turn the heat to medium-high. When the oil shimmers, add the onion and cook, stirring occasionally, until golden brown, about 10 minutes. Bundle the cooked onion with the bay leaf, thyme, and black peppercorns in a piece of cheesecloth and tie securely.
Wash the “lentils” thoroughly, then put them, the cheesecloth-wrapped aromatics, and the carrot in a medium saucepan. Add water to cover by about 3 inches. Bring to a boil and cook at a lively simmer, uncovered, until most of the water is absorbed, about 30 minutes. Turn the heat down to a minimum and cover. Cook another 10 minutes or so, then turn off the heat.
Discard the spice bag and the carrot, drain the lentils if necessary, and mash or semi-puree them, using an immersion blender, blender, or food processor. Season to taste with salt and white pepper. (The recipe may be made up to 1 day in advance to this point; cover and refrigerate.)
Warm the puree before serving, and serve warm or at room temperature, garnished with capers, shallots, chives, the remaining olive oil, and lemon juice.
