Make the polenta: In a large, heavy saucepan, combine the milk, water, bay leaf, thyme, garlic, salt, and pepper. Bring to a boil and slowly pour in the polenta, whisking constantly. Reduce the heat to low and simmer, stirring often, until the polenta thickens and the grains turn soft, about 35 to 45 minutes, depending on the variety. Remove the bay leaf and thyme sprigs and discard them. Turn off the heat and stir in the butter until it’s melted, then stir in the cheddar. Adjust to taste with salt and pepper. Keep the polenta warm while you grill the sausages and ramps.

Preheat a gas or charcoal grill, or place a cast iron grill pan over high heat. Cook the sausages on high heat at first, turning every 3 or 4 minutes, until all surfaces are browned. Move onto indirect heat (if grilling) or reduce stovetop heat to fully cook the sausages (around 10 to 15 minutes total). When done, move off the heat, and when they're cool enough, cut into diagonal slices.

Toss the ramps with 2 tablespoons of the olive oil and season them lightly with salt and pepper. Grill the ramps in a single layer until the leaves are wilted and lightly charred, and if you have bulbs, until they are tender, about 2 to 5 minutes, turning once.

Pour the polenta onto a serving platter and lay the grilled ramps and sausages on top. Drizzle with the last tablespoons of olive oil and serve hot with a little extra grated cheddar on top.