Arrange baguette slices in a single layer on a 12- by 15-inch baking sheet. Brush tops lightly with oil. Bake on the middle rack in a 350° regular or convection oven until golden, 15 to 20 minutes.
Meanwhile, trim and discard discolored stem ends, bits of debris, and bruised spots from mushrooms (for shiitakes, remove entire stem). Rinse mushrooms well and drain. Cut mushrooms larger than 1/2 inch into 1/4-inch-thick slices; leave smaller ones whole.
In a 10- to 12-inch frying pan over medium-high heat, melt butter; add garlic and stir often until fragrant, about 1 minute. Add mushrooms, shallots, and thyme; stir often until liquid is evaporated and mushrooms are well browned, about 10 minutes. Add wine, broth, and vinegar and stir to release browned bits; boil until liquid is evaporated, 4 to 6 minutes. Add salt and pepper to taste. Keep warm over low heat, stirring occasionally.
Spread chèvre equally on toasted baguette slices. Spoon warm mushroom mixture equally over cheese. Sprinkle evenly with parsley. Serve warm.