Position racks in the upper and lower thirds of the oven and heat the oven to 425°F. Bring a large pot of well-salted water to a boil over high heat.

Arrange the asparagus in a single layer on one half of a large rimmed baking sheet. Arrange the tomatoes on the other half of the sheet. Drizzle both with 1 Tbs. of the oil, season with 1/4 tsp. salt, and toss to coat. Roast on the top rack until the tomatoes have collapsed and the asparagus are bright green, about 20 minutes.

While the vegetables roast, put the almonds on another rimmed baking sheet and toast on the bottom rack, stirring occasionally, until fragrant and lightly browned, 10 to 12 minutes.

Reserve 1 heaping Tbs. of the almonds for garnish and put the remaining almonds in a food processor. Remove the tips from the asparagus and set aside. Roughly chop the remaining asparagus and add to the food processor along with the basil, cheese, 1/2 tsp. salt, and the remaining 3 Tbs. olive oil. Pulse until a coarse paste forms, about 10 seconds. Season to taste with salt and pepper and transfer to a large serving bowl.

Cook the linguine in the boiling water according to package directions until al dente. Drain the pasta and reserve 1/2 cup of the pasta cooking water. Add the pasta to the pesto and toss to coat. If necessary, add some or all of the reserved cooking water to loosen the pesto to a saucy consistency. Garnish with the tomatoes, asparagus tips, and reserved almonds, and serve