Place the eggs in a saucepan, cover with cold water by 1 inch, and bring to a gentle boil over medium-high heat. As soon as it boils, cover, remove from heat, and let stand 10 minutes. Transfer eggs to an ice bath until cool, then peel and quarter.
Cook the bacon in a large skillet over medium heat, stirring occasionally, until crisp and the fat has rendered, 8–10 minutes. Transfer bacon to a paper towel–lined plate. Pour off and reserve 3 tablespoons of the bacon fat in the skillet; discard any excess.
While the bacon cooks, prepare the vegetables. Add the baby spinach to a large mixing bowl. Add the red onion and cremini mushrooms on top.
Make the warm dressing: Set the skillet with 3 tablespoons bacon fat over low heat. Whisk in the red wine vinegar, Dijon mustard, granulated sugar, black pepper, and kosher salt. Cook, whisking, just until emulsified and steaming but not boiling, 30–60 seconds.
Pour the warm dressing over the spinach, onion, and mushrooms. Toss quickly and thoroughly until the leaves are glossy and just beginning to wilt, 20–30 seconds.
Add the crispy bacon and toss briefly to distribute. Arrange the quartered eggs over the salad and serve immediately while the greens are warm and the bacon is crisp.