Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high heat. Arrange the pierogies snugly flat-side down in the skillet (there may be some overlapping). Cook until the bottoms just begin to brown, about 3 minutes. Pour in 1/2 cup of water and simmer until the pierogi dough is tender and the water has completely evaporated, 4 to 5 minutes. Lower the heat to medium and continue cooking until the bottoms are golden brown, 2 to 3 minutes more; watch carefully as they can go from golden to burnt quickly. Remove from the heat.
While the pierogies cook, heat the remaining 1 tablespoon oil in a large skillet over medium heat. When the oil is shimmering, squeeze the sausage out of the casings into the skillet. Cook the sausage, stirring frequently and breaking it up into bite-size pieces, until browned, 5 to 6 minutes. Add the garlic and pepper flakes and stir until the garlic begins to brown, about 30 seconds. Stir in the broccoli. Stir in the broth, scraping the bottom of the pan to release any browned bits. Sprinkle with the Parmesan. Reduce the heat and simmer, uncovered, until the liquid is reduced by half and the broccoli is tender, 4 to 5 minutes. Remove from the heat.
Transfer the pierogies to a large, shallow serving dish and top with the sausage and broccoli mixture. Toss everything to combine and sprinkle with extra Parmesan.
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