Arrange a rack in center of oven; preheat to 300°. Pour 1 Tbsp. oil into center of a rimmed baking sheet, then arrange 1 lb. salmon over oil. Season salmon all over with salt and pepper, then drizzle with another 1 Tbsp. oil.
Roast salmon until flesh flakes in the thickest part of the fillet with pressed gently, 30–35 minutes. Meanwhile, fill a medium saucepan three-quarters full with water. Add several generous pinches of salt and bring to a boil, covered, over medium-high heat.
While salmon is cooking and water is coming to a boil, focus on the rest of your salad prep. First, trim stem end off of 8 oz. green beans.
Cut 1 cup tomatoes in half.
Peel and finely chop 1 shallot. Transfer to a medium bowl; set aside.
Separate leaves from 1 head of lettuce and arrange across a large platter. Cover with damp paper towels and chill until ready to serve.
When water is boiling, add green beans and cook until tender but still retain some snap, about 2 minutes. As you wait for the green beans to cook, fill another medium bowl halfway with ice water. Using tongs, remove green beans immediately (but leave boiling water right where it is) and transfer to bowl with ice water. Leave green beans there for 15–20 seconds, or just long enough to chill them completely, then remove from water and pat dry; set aside. Keep the ice water for cooking the eggs.
Add 4 eggs to saucepan and cook exactly 8 minutes. Transfer to ice water bath and let cool 2 minutes. Peel eggs, keeping them submerged in ice bath as you work. Pat eggs dry and transfer to cutting board.
Time to make your dressing: Add 1 Tbsp. mustard, 1 Tbsp. vinegar, and a pinch of sugarto bowl with shallot; season with salt and a generous amount of pepper. Whisking constantly, slowly stream in remaining 6 Tbsp. oil until you have a thick, emulsified dressing.
Remove platter with lettuce from fridge and drizzle with about one-third of dressing. Arrange green beans, tomatoes, and ½ cup olives over lettuce. Cut eggs in half and tuck into gaps on platter.
Using your hands or a fork, flake off large pieces of salmon and arrange over salad.
Drizzle platter with remaining dressing. Season with salt and pepper.