Melt butter in a heavy large saucepan over medium heat. Add milk and bring to a simmer. Whisk in processed cheese. Gradually whisk in shredded cheeses. Stir until cheeses melt and mixture begins to bubble, about 5 minutes. Whisk in cayenne.

Meanwhile, cook the tortellini in a pot of boiling, salted water until just tender, about 4 minutes. Drain. Add tortellini to cheese sauce. Toss to coat. Divide tortellini and sauce equally among the pasta bowls and serve immediately.