Soak the chickpeas: a large bowl, dissolve 1 tablespoon of kosher salt in the water. Add the chickpeas and soak at room temperature for 8 to 24 hours. Drain and rinse.

Sauté the vegetables: Preheat the Instant Pot by selecting Sauté and adjust to More for high heat. Add the olive oil and heat until it shimmers. Add the onion and sprinkle with 1/4 teaspoon of salt. Cook, stirring frequently, until the onion pieces separate and soften, 2 to 3 minutes.

Pressure cook: Add the drained chickpeas, tomatoes with their juice, stock, and 1/4 teaspoon of salt. Lock the lid into place. Select Manual; adjust the pressure to High and the time to 15 minutes. After cooking, naturally release the pressure for 10 minutes, then quick release any remaining pressure. Unlock and remove the lid.

Serve the chickpeas: Ladle the chickpeas into bowls and top each with a spoonful of pesto. Sprinkle with the cheese and serve.