Cook the eggs: Bring a small pot of water to a boil. Carefully add the eggs and cook for 7 minutes for jammy centers. Transfer immediately to an ice bath and let cool completely before peeling.
Make the vinaigrette: In a medium bowl, whisk together the whole grain Dijon, lemon juice, olive oil, dill, salt, and pepper.
Prepare the egg salad: Roughly chop the eggs and fold them into the vinaigrette. The mixture should be chunky with visible pieces of jammy yolk. Set aside.
Season the tuna: Pat the tuna dry and season all sides with the salt. Press the cracked black pepper evenly over the exterior. (see note)
Sear the tuna: Heat the neutral oil in a skillet over high heat until shimmering. Sear the tuna for about 30–45 seconds per side, including the edges, until a thin crust forms while the center remains rare. Transfer to a cutting board and let rest for 2 minutes before slicing thinly. Make sure to not wait longer than 2 minutes as the tuna will continue to cook.
Toast the bread: Toast the sourdough halves until golden and crisp.
Assemble: Spoon the egg salad evenly over each toast. Top with slices of seared tuna and garnish with additional dill, flaky salt, and a squeeze of fresh lemon juice.
Serve: Enjoy immediately while the toast is crisp and the tuna is freshly seared.