How to Make It
In a food processor, pulse the 3 1/2 cups of flour with the salt. Add the eggs and water and pulse until the dough starts to come together. Turn the dough out onto a work surface and knead by hand until smooth and elastic, about 10 minutes. If the dough is too sticky to work with, lightly dust it with flour. Wrap the dough in plastic and let rest at room temperature for 1 hour.
In a large saucepan, melt the butter. Add the onion, rosemary, thyme, garlic and a pinch of salt and cook over moderate heat, stirring occasionally, until the onion is softened but not browned, about 8 minutes. Add the flour and cook, stirring constantly, until the roux is light golden, 3 to 5 minutes. Gradually whisk in the milk and bring to a boil, then simmer over moderately low heat, stirring frequently, until no floury taste remains, about 20 minutes. Press the béchamel through a fine sieve into a bowl; discard the solids. Season with salt and let cool.
Cut the dough into 8 equal pieces; work with 1 piece at a time, keeping the rest covered with a towel. Flatten the dough slightly. Run it through a pasta machine a total of 6 times: Start at the widest setting, then run through successively narrower settings. Dust the sheet with flour and lay it on a parchment paper-lined baking sheet. Repeat with the remaining dough, separating the sheets with parchment.
In a large pot of salted boiling water, cook the pasta sheets until just al dente,
1 to 2 minutes. Drain in a colander and cool under running water, then drain again. Return the pasta to the baking sheet and toss with olive oil to prevent the sheets from sticking together.
Preheat the oven to 350°. Brush a deep 9-by-13-inch baking dish with oil; spread with 1/2 cup of the béchamel. Arrange a layer of pasta over the béchamel, trimming to fit. Spread one-fifth of the remaining béchamel over the pasta; sprinkle with 1/2 cup of the Fontina and 1/4 cup of the Grana Padano. Repeat the layering 4 more times, ending with the cheeses.
Tightly cover the baking dish with foil and bake the lasagna for 45 minutes, until bubbling. Remove from the oven and uncover. Preheat the broiler. Broil the lasagna 6 inches from the heat until lightly browned on top, 2 to 4 minutes. Let rest for 15 minutes, then cut into squares and serve with one of the toppings.