Step 1Make fresh pasta dough by pulsing the flour and 4 teaspoons olive oil in a food processor. Add the egg and pulse until the dough forms a ball that rides around on the top of the blade. Remove the dough from the food processor and knead until smooth and shiny; this should take only about a minute. Wrap in plastic wrap and refrigerate at least 1 hour.
Step 2Roll the pasta dough out very thin, about a 6 setting on most machines, flouring as necessary. The sheets should be at least 4 inches wide. Cut the pasta sheets into squares, dust lightly with flour and set aside until ready to use.
Step 3Cook four or five pasta sheets at a time in plenty of rapidly boiling salted water. The pasta will be done when sheets float to the surface, 1 to 2 minutes. Remove from the boiling water and drain on a tea towel if using immediately or transfer to a large bowl of water to store. Repeat, using all of the pasta.
Step 4Heat the oven to 375 degrees. Place the tomato slices on a rack, drizzle with one-fourth cup olive oil, and season with one-fourth teaspoon salt and a couple grinds of black pepper. Roast the tomatoes for 40 minutes, until the surface is lightly caramelized and the tomatoes are softened. Remove from the oven and allow to cool slightly.
Step 5While the tomatoes are cooking, beat together the ricotta, one-fourth teaspoon salt, minced garlic and parsley in a small bowl until thoroughly mixed. Set aside.
Step 6To make the pesto, drop the garlic cloves through the feed tube of a running food processor or blender and mince until fine. Turn the machine off and add the basil leaves. With the machine running again, slowly pour the remaining one-fourth cup olive oil through the food tube. Add the pine nuts and pecorino Romano and pulse to combine and season to taste with salt. (The dish can be prepared to this point several hours in advance. The tomatoes and the pesto can be held at room temperature or refrigerated; the ricotta should be refrigerated; and the cooked pasta sheets should be stored in a large bowl of water. Bring the pesto and ricotta to room temperature, warm the tomatoes in a skillet over low heat and drain the pasta sheets and pat dry on a tea towel before using.)
Step 7Heat the oven to 350 degrees. Place one square of the cooked pasta on a heat-proof plate and brush on enough of the pesto to thinly coat the sheet. Spoon one-fourth cup of ricotta in a low mound in the center. Arrange about three slices of roasted tomatoes on top of the ricotta. Brush one more pasta square with the pesto, and place this on top of the tomatoes. Repeat with three more plates.
Step 8Place the plates on a cookie sheet and place in the oven to heat through, about 8 minutes. Serve immediately.