Trim root end of 1½ pounds leeks and remove tough dark-green ends. Halve lengthwise and cut into ½-inch pieces. Wash and drain thoroughly. Transfer to bowl and set aside.
Remove stalks and fronds from 1 fennel bulb. Halve bulb, remove core, and cut into ¼-inch pieces. Add to bowl with leeks.
Halve 1½ pounds kabocha squash. Scrape out seeds, then peel and cut squash into 1-inch pieces. Set aside.
Cook 1 pound hot Italian sausage in Dutch oven over medium-high heat until browned, 5 to 7 minutes, breaking up meat with wooden spoon.
Use slotted spoon to transfer sausage to plate, leaving fat behind in pot; set sausage aside.
Add leeks, fennel, 4 thinly sliced garlic cloves, 1 tablespoon extra-virgin olive oil, ¼ teaspoon table salt, 1 bay leaf, and 4 thyme sprigs to pot. Reduce heat to medium and cook, stirring occasionally, until vegetables are tender and just beginning to brown, about 7 minutes.
Stir in squash; 6 cups chicken broth; 1 (15-ounce) can cannellini beans, rinsed; and sausage, scraping up any browned bits. Cover pot, increase heat to high, and bring soup to vigorous simmer.
Reduce heat to medium and simmer vigorously until squash is tender, 20 to 25 minutes, adjusting heat as needed.
