Preheat oven to 425° and line large rimmed baking sheet with foil. Place shallots on a prepared baking sheet and drizzle with oil; toss to coat. Roast shallots until they are lightly browned; 12 minutes.
Next, add asparagus, mushrooms, vinegar, thyme, pepper, and ¼ tsp salt, and 1 tbsp oil to the pan with the shallots; stir to combine. Roast another 20 minutes, until veggies are tender.
Meanwhile, combine milk and stock in a large saucepan; bring to a boil over medium-high heat. Whisk in the polenta; reduce heat to medium-low and cook until thickened, whisking often; about 5 minutes.
Remove from the heat and stir in the parmesan. Then, heat the remaining 1 tbsp of oil in a large nonstick skillet over medium heat. Add eggs 1 at a time and cook until whites are cooked through, but yolk is not; about 3 minutes.
To serve, spoon polenta into bowls and top with veggeis and egg. Add more salt if desired!