Finely grate half of the Parmesan into a large bowl. Add shallot, vinegar, honey, and 5 Tbsp. oil and whisk well; season dressing with salt and pepper.
Cook bacon in a large skillet over medium-high heat, stirring occasionally, until brown and beginning to crisp, about 5 minutes. Add rosemary and cook, turning once, until crisp, about 2 minutes (fresh rosemary can be intense; frying the sprigs mellows their flavor and creates a crisp salad topper). Using a slotted spoon, transfer bacon and rosemary to paper towels.
Add remaining 2 Tbsp. oil to skillet and heat over medium-high. Arrange mushrooms in pan in a single layer and cook, undisturbed, until golden brown underneath, about 3 minutes. Season with salt and pepper, toss, and continue to cook, tossing often, until golden brown all over, about 5 minutes. Transfer to bowl with dressing, but don’t toss. Strip rosemary leaves off stems into bowl and add radicchio.
Meanwhile, bring a large pot of water to a boil. Reduce heat so water is at a bare simmer. Crack an egg into a small bowl; gently slide egg into water. Quickly repeat with remaining eggs. Poach, rotating eggs gently with a large slotted spoon, until whites are set but yolks are still runny, about 3 minutes. Using spoon, transfer eggs to paper towels. Season with salt and pepper.
Toss salad to coat leaves; season with salt and divide among plates. Shave remaining Parmesan over and top with bacon and poached eggs.