Heat 1 tablespoon of the oil in a medium saucepan over medium heat until shimmering. Add the onion and garlic, season with salt and pepper, and cook, stirring occasionally, until softened, about 5 minutes.
Add the lentils, stir to combine, and add the chicken broth. Increase the heat to high and bring to a simmer. Reduce the heat to low and simmer, stirring occasionally, until the lentils are just tender and the water has evaporated, about 30 minutes. Remove from the heat and set aside.
Meanwhile, trim the ends from the chard stems and discard. Cut off the stems at the base of the leaves and slice the stems crosswise into 1/4-inch pieces. Place in a small bowl and set aside. Stack the leaves, cut them in half lengthwise, then coarsely chop into bite-sized pieces; set aside.
Heat the remaining tablespoon of oil in a large frying or straight-sided pan over medium-high heat until shimmering. Add the reserved chard stems, season with salt and pepper, and cook, stirring occasionally, until softened, about 5 minutes.
Add the chopped chard leaves, measured salt, and measured pepper and cook, stirring occasionally, until wilted, about 2 minutes. Stir in the red wine vinegar and reserved lentil mixture until evenly combined. Remove from the heat and allow to cool slightly, about 3 minutes.
Sprinkle in the feta and stir to combine. Taste and season with salt and pepper as needed. Serve warm or at room temperature.