In a small dish place the dried porcini mushrooms, soak them in warm water for 20 to 30 minutes. Remove from water, squeeze the water out and roughly chop. Save the soaking water and add the liquid to the mushroom stock. Set the porcini aside.
Over medium-high heat, pour 3 tablespoons olive oil into a heavy-bottomed large Dutch oven pan
Cook the mushrooms in batches by adding half of the mushrooms with half of the pearl onions, with ¼ teaspoon salt and ¼ teaspoon freshly ground black pepper to the pan. Cook covered for about 6 minutes until all the liquid has rendered, and the mushrooms are tender and caramelized. Remove from the heat and set aside. Repeat the same process with the remaining half of mushrooms and pearl onions.
Once the second batch is done, uncover and add the first batch of mushrooms back to the pan, reducing the heat to low. Add the minced garlic and continue to cook until mushrooms are tender about 8 minutes.
When done, transfer the mushrooms mixture to a bowl and set aside.
Return the pot to the stove over medium-high heat and add the remaining 2 tablespoons olive oil and flour by carefully whisking the flour and oil until well combined and the combination starts to get a light caramel color.
Add the carrots, onions, garlic cloves, anchovy paste, tomato paste, and soaked and chopped porcini mushrooms. Stir well and cook until the onions are translucent about 8 to 10 minutes.
Meanwhile, in a small bowl dissolve the agar-agar or corn starch with ½ cup of the vegetable broth. Set aside.
When the carrots and the onions are done add the corn starch-broth combination and stir well.
Add the wine to the center of the pan to deglaze by stirring and dislodging any brown bits that may be stuck to the bottom or the side of the pan.
Carefully pour in the mushrooms stock to which you added the porcini mushrooms water, fresh thyme, bay leaves, and peppercorns. Bring to a boil and cook while stirring occasionally and the liquid has thickened somewhat. About 10 to 12 minutes.
When the liquid is thickened, it should be slowly coming off the spoon almost like a light syrup consistency, remove from heat.
Over a large bowl, place a large sieve and place the pan vegetables with the liquid through the sieve to separate the liquid from the solids. With a rubber spatula, press the solids to get as much juice as you possibly can.
Return the liquid to the pan and discard the solids. If at this point if your sauce is too thin add equal parts liquid and corn starch and mix them into the liquids and let it thicken by stirring well.
Season with salt and pepper. Be careful when adding the salt as there is already 1 tablespoon anchovy paste which should not be omitted as it gives the dish great depth of flavor.
Lower the heat to medium-low and add the mushrooms/pearl onions mixture and cook until heated thru about 5 minutes.
Serve over creamy polenta, pasta, or silky-smooth potato puree. Sprinkle some finely chopped parsley over your dish, optional.