Cut the rabbit into pieces. Rinse under cold water and pat dry with paper towels. Roughly chop half of the rosemary sprigs (about 1 sprig), the thyme sprig, and the garlic cloves. Mix with olive oil, salt, and pepper. Coat the rabbit pieces thoroughly with this mixture, cover, and refrigerate for 3 to 4 hours.
Heat olive oil in a large roasting pan over medium-high heat. Brown the rabbit pieces well on all sides, about 8-10 minutes total. Add the finely chopped onion and continue cooking. Deglaze by adding roughly half a glass of water, scraping up the browned bits from the bottom of the pan. Continue cooking, adding more water in similar amounts as it evaporates, and stir frequently. Cook until the onion has turned a deep brown color, about 15-20 minutes total. Keep the mixture moist without letting it burn.
While the onion cooks, finely chop the carrots and celery. Once the onion is deep brown, add the carrots, celery, the remaining rosemary sprig, and the bay leaves. Sauté for 1 minute. Stir in the tomato paste and cook for 1-2 minutes. Pour in the broth, stirring well, and let it reduce until almost completely evaporated, about 5-10 minutes. Pour in the full bottle of Valpolicella wine, stir well, and gently simmer with the lid on for 20 minutes.
Remove the rabbit pieces, bay leaves, and rosemary sprigs from the pan and set aside. Using an immersion blender, purée the sauce until very smooth. Season with salt and pepper to taste. Return the rabbit pieces to the pan and coat thoroughly with the sauce. Transfer to a preheated oven at 180°C (356°F) and heat for 20 minutes.
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