Combine all meatball ingredients in a bowl and mix until incorporated. Do not pack the meat as you mix. Form into 1½-inch meatballs. Heat olive oil in a large saucepan and cook meatballs over moderately high heat, turning until browned all over. Alternatively, bake meatballs in an oven at 375°F for 20-25 minutes.
For sauce, heat olive oil in another saucepan and add onion, carrot, garlic, and a generous pinch of salt. Cook over medium heat, stirring occasionally, until just beginning to brown. Add tomatoes, paprika and red pepper flakes. Simmer over moderate heat until slightly reduced, 3-5 minutes. Stir in tomato paste, wine and sherry and simmer until barely thickened. Transfer to a blender and purée until very smooth. Return to saucepan, whisk in stock and bring to a boil. Season with salt and fresh cracked pepper.
For mushrooms, heat oil and butter in a pan over medium-high heat, then add the mushrooms, thyme sprigs and garlic. Give a pinch of salt and sauté until mushrooms have crispy edges. Don't crowd the pan. If mushrooms are large, cut them into bite-size pieces.
Spoon meatballs and sauce into bowls, top with mushrooms and a pinch of chopped parsley and serve.