For the Mushrooms:
Heat a large skillet over medium-high heat and melt butter. Once melted, add mushrooms (working in batches if needed to not overcrowd the pan), garlic and thyme and cook, tossing occasionally, until mushrooms are lightly browned and tender. If working in batches, remove mushrooms from skillet and set aside, keeping the garlic and thyme in the pan and add a bit more butter to the pan, and repeat with remaining mushrooms.
Add shallots, season everything with salt and pepper, and saute for 1-2 minutes, until shallots are tender and fragrant. Discard garlic and thyme, then add a splash of sherry vinegar, crème fraîche then stir to combine. Reduce heat to medium low and let simmer all together for a moment. Taste and adjust seasoning before removing from heat.
Place a few generous spoonfuls of mushrooms and sauce on top of toasted bread, then top with arugula leaves. Nestle the eggs atop the greens and top with shaves of Gruyère.