Place a large high-sided skillet over medium-high heat. Combine the milk, 3 1/2 cups water and a large pinch of salt in a medium saucepan, cover and set it over medium-high heat.
When the large skillet is hot, add 2 tablespoons of the oil and swirl the pan to coat. Add the sausage and cook, using a wooden spoon to break up the meat, until lightly browned in parts, about 3 minutes. Meanwhile, cut the onion into large chunks.
Add the onion, carrots and garlic to a food processor and pulse until finely chopped. Add 2 tablespoons of olive oil, the vegetables and a large pinch of salt to the pan with the sausage and cook until the vegetables are slightly softened, about 3 minutes.
By this time, the milk and water mixture should be at a strong simmer. Whisk in the polenta and lower the heat to medium-low. Cook, stirring frequently, until the polenta is thickened and creamy, about 3 minutes. Stir in the butter and cheese and season with additional salt if needed. Remove from the heat and cover to keep warm.
When the vegetables are slightly softened, stir in the tomatoes and red pepper flakes. Cook, uncovered, stirring occasionally, until the vegetables are completely tender and the juice from the tomatoes has reduced slightly, 6 to 8 minutes. Season to taste with additional salt and red pepper flakes if desired.
Divide the polenta (if the polenta is too thick, whisk in 1/4 to 1/2 cup warm water to thin it out) among 4 bowls and top with the chunky tomato ragu. Toss the greens in a large bowl with the remaining olive oil, a squeeze of lemon juice, and some salt and pepper and serve on the side.
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