How to Make It
In a large skillet, heat 1/4 cup of the olive oil. Add the shallot and cook over high heat, stirring, until fragrant, about 30 seconds. Add the mushrooms and thyme, season with salt and pepper and cook, stirring occasionally, until browned, 8 to 10 minutes. Add the sherry and cook until evaporated, about 1 minute. Remove from the heat.
In a large bowl, whisk the mustard with the vinegar. Gradually whisk in the remaining 1/4 cup of olive oil and season with salt and pepper. Stir in the mushrooms and parsley and refrigerate until the mushrooms are cool, about 15 minutes.
Add the pea shoots and Parmigiano-Reggiano shavings to the mushrooms and toss, then serve.