1. Combine 2 tablespoons water, shallot, and ½ teaspoon salt in large saucepan. Cover and cook over medium heat until shallot begins to soften, about 3 minutes. Uncover, stir in garam masala and ⅛ teaspoon pepper, and cook until fragrant, about 1 minute.
2. Stir in fresh and dried apricots, sugar, and remaining 1½ cups water. Cover and simmer, stirring occasionally, until apricots are just softened, 10 to 15 minutes.
3. Stir in lemon zest and lemon juice. Increase heat to medium-high and simmer uncovered until mixture is thickened, 5 to 8 minutes. Season with salt, pepper, and sugar to taste, and let cool slightly.
4. Using funnel and spoon, portion chutney into two 1-cup jars. Let cool to room temperature and serve. (Chutney can be refrigerated for up to 2 months.)