For the chutney. In a small sauce pot on medium heat, add the preserves, giardiniera, bell pepper, vinegar, a pinch of salt and a few grinds of pepper. Add a splash of water to loosen and bring to a simmer. Simmer, stirring occasionally, until the bell pepper becomes tender, about 10 minutes. Remove from the heat and stir in the parsley and mint.
For the green beans. Prepare a grill for high heat and place an inverted wire resting rack on top of the grill.
Place the green beans in a single layer on top of the wire rack and close the grill lid. Grill, rotating midway through, until bright green, 5 to 8 minutes.
Transfer the grilled green beans to a large bowl, toss with the chutney, season with salt and pepper and serve.