Preheat the oven to 200 degrees . Heat a grill pan over medium-high. Brush the mushroom tops with the oil and grill until softened and charred in spots, about 5 minutes per side; season. Transfer to a baking sheet and keep warm in the oven.
In a large pot, toss the spinach over medium heat until just wilted, 3 to 5 minutes; drain. Stir in the lemon juice, 1 tbsp. butter and 1/4 tsp. salt; drain.
In a small saucepan, melt 1 tbsp. butter over medium. Add the flour; whisk until golden, about 1 minute. Whisk in the milk. Cook, whisking often, until thickened, about 3 minutes. Add the cheese and 1/4 tsp. salt; whisk until smooth. Cover and reduce heat to low; stir occasionally.
In a large skillet, bring 3 inches water to a boil; reduce heat to low. Crack eggs into the skillet, spacing apart. Poach until whites are just set, 3 to 4 minutes.
Place 2 mushrooms upside-down on each plate and top with spinach mixture. Use a slotted spoon to transfer eggs to each and top with sauce. Garnish with the pine nuts, lemon zest and pepper.