Combine spinach, eggs, and mushrooms in a large shallow serving bowl.
Cut bacon on the diagonal into strips about 1/4 in. thick and 1 1/2 in. long. Cook bacon with 1/3 cup water in a large frying pan over medium heat, stirring occasionally, until water disappears, 8 to 12 minutes. Add oils and cook until bacon is light golden but still supple, 3 to 5 minutes more. Transfer bacon with a slotted spoon to paper towels to drain.
Add fennel to pan and cook, stirring occasionally, until slightly softened, about 2 minutes. Transfer with a slotted spoon to more paper towels.
Stir shallot and thyme into fat in pan and cook until softened, about 2 minutes. Remove pan from heat and whisk in vinegar to deglaze pan. Whisk in mustard, 1/2 tsp. each salt and pepper, and 2 tsp. water.
Add bacon and fennel to salad, scrape dressing on top, and toss gently to coat. Season to taste with more salt and pepper.
Note: Nutritional analysis is per serving.