Step 1Bring a large pot of water to a boil, add the vegetable oil and cook the pasta to al dente following the manufacturer's instructions. Drain the pasta, spreading it out onto a rimmed baking sheet, set aside while preparing the rest of the recipe.
Step 2In the same pot, heat the olive oil over medium heat. Add the bacon and cook, stirring occasionally, until the bacon is crisp and the fat is rendered. Strain the bacon from the fat in the pan, and drain the excess fat, leaving only 3 tablespoons fat in the pot. Stir in the garlic, then the greens, a handful at a time. Continue stirring until all of the greens have been added and are wilted.
Step 3Stir the bacon back in with the greens, then the pasta, tossing to evenly combine the pasta and flavorings. Season with a pinch of salt and several grinds of black pepper.
Step 4Divide the pasta among warmed shallow bowls, topping each portion with a squeeze of lemon juice. Garnish each plate with Parmigiano-Reggiano and serve immediately.