In a small bowl, add the jalapeño and 1 tablespoon of the vinegar. Set aside to pickle for 5 minutes.
In a medium saucepan set over medium heat, add the mushrooms, chicken broth, olive oil, shallot, thyme and 1 teaspoon kosher salt. Cover and cook until the mushrooms are softened, about 5 minutes. Remove the lid and stir in the lemon juice. Season with additional salt if needed.
In a medium saucepan set over medium heat, add the remaining 1 teaspoon vinegar and enough water to reach 3 inches up the sides of the pan. Bring to a boil. Working one at a time, crack each egg into a small bowl and carefully tip the egg into the boiling water. Repeat with another egg. Use a slotted spoon to baste the eggs, cooking until the whites are set, about 2 minutes. Use the slotted spoon to carefully transfer the eggs, one at a time, to a paper-towel-lined plate, then repeat with the remaining eggs.
Divide the mushrooms and broth among 4 bowls. Place 1 egg in each of the bowls and season each serving with salt and pepper. Sprinkle the Parmigiano-Reggiano cheese around each egg and top with a few of the pickled jalapeño slices and the parsley. Serve immediately. Calories per Serving: 120; Sodium: 710mg; Total Carbohydrate: 4g; Fiber: 1g; Fat: 7g