How to Make It
Preheat oven to 325°F. Spread almonds in an even layer on a rimmed baking sheet. Bake in preheated oven until toasted, 8 to 10 minutes, stirring once after 5 minutes. Let cool 5 minutes.
Coarsely chop almonds. Place almonds, 2 tablespoons extra-virgin olive oil, garlic, lemon zest, and 1/4 teaspoon kosher salt in a food processor. Pulse until mixture is finely chopped, about 12 pulses. Transfer mixture to a small bowl. Stir in remaining 2 tablespoons extra-virgin olive oil; add kosher salt to taste. Let stand at room temperature at least 30 minutes.
Score the cut sides of each mushroom half with a 1/8-inch-deep diamond pattern. Heat 3 tablespoons olive oil in a large cast-iron skillet over medium-high. Place 6 mushroom halves, cut sides down, in hot oil. Sprinkle with 1/2 teaspoon kosher salt and 1/4 teaspoon pepper. Wrap the bottom of a second large cast-iron skillet with aluminum foil, and place on top of mushrooms. (If using a lighter skillet, weight down with canned goods.) Reduce heat to medium; cook mushrooms, weighted down with skillet, until golden brown, 5 to 7 minutes. Remove top skillet; flip mushrooms, and sprinkle with 1/4 teaspoon kosher salt and 1/4 teaspoon pepper. Continue cooking mushrooms, weighted down with skillet, until tender and golden on both sides, 4 to 6 minutes. Remove mushrooms from skillet. Return skillet to medium-high, and repeat process with remaining olive oil, mushrooms, kosher salt, and pepper.
To serve, spoon about 3 tablespoons pesto on each of 6 plates. Top evenly with mushroom halves and almond aillade. Drizzle with lemon juice; sprinkle with flaky sea salt to taste, and garnish with additional lemon zest.