Bring the chicken broth to a boil in a medium saucepan. Reduce the heat to medium low, stir in the oats and cook, stirring occasionally, until tender and thickened, about 20 minutes.
Meanwhile, cook the prosciutto in a large skillet over medium-high heat, flipping once, until crisp, 4 to 5 minutes. Transfer to a paper towel¿lined plate to drain.
Add the olive oil, garlic, 1 teaspoon thyme and the rosemary to the skillet. Reduce the heat to medium and cook until the garlic is lightly toasted, 10 seconds. Stir in the tomatoes and 1/2 cup water. Simmer, stirring occasionally, until the sauce is slightly thickened, 6 to 8 minutes. Transfer to a medium bowl and season with salt and pepper; cover to keep warm.
Wipe out the skillet and fill halfway with water; bring to a gentle simmer over medium-high heat. Carefully crack the eggs into the water and cook until the whites are set but the yolks are still runny, 2 to 3 minutes. Transfer to a plate with a slotted spoon and season with salt and pepper. Top each serving of oats with the tomato sauce and parmesan. Sprinkle with the remaining 1 teaspoon thyme and top with the poached eggs and prosciutto.