Butter an oven-proof pan. Dust the pan all over with 1/3 cup Parmesan. Melt 1/4 cup butter in a large saucepan over medium heat. Add flour: whisk for 2 minutes. Gradually whisk in milk. Bring to a simmer, whisking often. Reduce heat to medium. Cook, whisking frequently, until sauce is thickened, about 20 minutes. Remove from heat. Whisk in Fontina and 1 cup Parmesan. Add egg; whisk to blend. Season with salt and pepper. Cover sauce and keep warm.
Arrange a rack in upper third of oven; preheat to 375°. Cook pasta in a large pot of boiling salted water, stirring occasionally, until almost al dente (~6-7 minutes). Using a large slotted spoon, transfer the pasta to a large bowl. Add veggie of choice to pasta water; cook until just tender, about 1 minute. Drain; add to bowl with pasta. Stir in cheese sauce. Add arugula, 1 cup parsley, and 1/2 cup mint to pasta mixture; mix to evenly incorporate.
Transfer half of pasta mixture to prepared pan. Mix ricotta and lemon zest in a small bowl; dot half of ricotta mixture over pasta. Spoon remaining pasta mixture into pan and dot with remaining ricotta mixture. Sprinkle with 1/4 cup Parmesan.
Bake pasta for 30 minutes. Sprinkle with remaining 1/4 cup Parmesan. Bake pasta until top is golden brown, about 10 minutes longer. Let rest for 30 minutes; remove pan sides. Sprinkle with 2 tbs of parsley and 2 tbs of mint and cut into wedges.