Cook the mushrooms: Add oil, garlic, thyme, mushrooms, and salt to the largest skillet you have (I like to use a 12-inch cast-iron skillet). Turn the heat on to medium-high and cook, stirring occasionally, until the mushrooms are tender and golden brown. Deglaze with sherry vinegar, season to taste with salt, discard garlic and thyme. Cool to room temperature before transferring to an airtight container and refrigerating.
Cook the eggs: Beat together the eggs and salt in a small mixing bowl until homogenous and frothy.
Add the olive oil to a medium nonstick skillet, heat for about 30 seconds over medium-high heat, then add the mushrooms and sun-dried tomatoes, and cook for 1 minute, stirring occasionally. (This is not to cook the vegetables but rather to heat them up.) Once the pan and vegetables are hot, add the egg and cook, stirring constantly for 30 seconds to 1 minute or until the egg looks not quite soft-cooked. (The residual heat from the skillet will continue to cook the eggs.) Remove from the heat, add the goat cheese, stir to combine, and transfer to a plate. Finish with pepper and a sprinkling of parsley. Serve with toast.