In a medium saucepan, combine the potatoes and 1 tablespoon salt, then add water to cover by 2 inches.
Bring to a boil over medium-high, reduce to medium and cook, uncovered and stirring occasionally, until the potatoes are just shy of tender, about 3 minutes; adjust the heat as needed to maintain a simmer.
Add the cauliflower and beans; cook, stirring occasionally, until the vegetables are just tender, 3 to 5 minutes.
Drain in a colander, immediately add to the bowl with the chili puree and toss until well coated.