Heat the lobster stock in a saucepan over medium heat. Add the lobster tails and cook for 10 to 12 minutes. Remove to a sheet tray and set aside to cool. Reserve the stock for later. Remove the meat from the shells and cut into nice bite-sized pieces.
To a large skillet, add the mushrooms and cook, stirring occasionally, until brown, 8 to 10 minutes. Add a drizzle of olive oil and some salt. Remove from the heat. Dice half of the sauteed mushrooms and leave the other half whole for garnish.
Heat a large high-sided saute pan over medium-high heat with some olive oil. Add the onion and some salt and sweat until translucent, 5 to 7 minutes. Add the farro and stir to coat in the onions and olive oil. Toast for 5 minutes, then add the wine. Season with salt, then allow the wine to reduce, stirring occasionally, until it's almost completely absorbed. Add the tomato paste and stir to combine. Begin to add the stock 2 ladlefuls at a time. Cook, stirring frequently, until the stock has absorbed into the farro. Repeat adding the stock two more times until the farro is tender but still has some bite to it. (There may be leftover stock.) Stir in the zucchini. Add the butter and Parmesan and stir vigorously to combine. The farro should be loose and creamy. Fold in the diced lobster mushrooms, diced lobster meat and chives.
Plate the farrotto and garnish each portion with a slice of lobster mushroom. Garnish with the micro basil.