In a large pot fitted with a steamer basket, bring about 1/2 inch of water to boil over high heat. Add the Broccolini, cover, and steam until just tender, 6 to 7 minutes. Transfer to a large towel-lined baking sheet and let cool briefly; then pat very dry with another towel.
In a 12-inch skillet, cook the bacon over medium heat, stirring frequently, until golden, about 5 minutes. Use a slotted spoon to transfer the bacon to a small bowl, leaving the fat in the pan.
Increase the heat to medium high and arrange half of the Broccolini in a snug single layer in the skillet, making sure all have good contact with the bottom of the pan. Sprinkle with 1/4 tsp. salt. Let cook undisturbed until browned in spots, about 1-1/2 minutes. Transfer to a serving platter. Add 1 Tbs. of the olive oil to the skillet and sear the remaining Broccolini, sprinkling it with another 1/4 tsp. salt.
Return the first batch of Broccolini to the skillet. Sprinkle with the bacon, lemon juice, chile, and the remaining 1 Tbs. of olive oil. Toss well to combine and remove from the heat. Sprinkle with the Parmigiano, quickly toss to combine, and transfer to the serving platter. Serve immediately.
The Broccolini may be steamed up to 2 hours ahead.