Preheat the oven to 350°F.
In a large pot (Le Creuset pot works well) over medium high heat, add the sausage and fry, stirring occasionally, until golden. Using a slotted spoon, remove the sausage from the pot and transfer to a paper towel-lined plate. Using the same pot and the fat from the sausage, fry the onion and garlic over medium heat until the onion is translucent and the garlic is fragrant, about 5 minutes. Add the leek, celery and carrot and continue to sauté, stirring occasionally, for about 5 minutes. Add the spinach and cook until just wilted. Add in the beans, bay leaf, thyme, sausage, and broth. In a separate small bowl, whisk the cornstarch with 3 tablespoons of water and whisk into the pot. Bring to a simmer and cook, uncovered, until the vegetables are tender, about 15 minutes. At this point, the liquid should be thickened. If you find it to be too thin, continue to simmer for a few minutes longer.
While the mixture simmers, make the biscuit topping: In a large bowl, mix together the flour, baking powder, and salt. Cut the butter into the flour mixture using a pastry cutter or your fingers until you have a coarse crumb mixture. Add the cheese and then mix in the milk with a fork until a sticky dough forms. Set aside.
Once thickened, pour the sausage mixture into a 9 x 13 inch baking dish. Top with large spoonfuls of the dough and transfer to the oven. Bake until the biscuit topping is nicely browned and the filling below is bubbling up the sides, about 45 minutes. Air on the side of baking for longer even if the filling is bubbling before 45 minutes to ensure all of the biscuits are cooked through.