Tip:To grill green beans, simply arrange the beans perpendicular to the grates. To flip, use a shoveling motion with your tongs to scoop up a few and turn them over.
Prepare a medium-high gas or charcoal grill fire. In a small bowl, combine 1 Tbs. of the oil, 1 Tbs. of the mustard, the brown sugar, 1-1/2 tsp. salt, and 1/2 tsp. pepper. Brush all over the steak and set aside.
Brush the green beans, scallions, corn, and tomatoes all over with the remaining 1-1/2 Tbs. oil and season with 1-1/2 tsp. salt and 1/2 tsp. pepper. Grill the vegetables, turning occasionally, until softened and lightly charred, 1 to 2 minutes for the scallions, 3 to 5 minutes for the beans, 4 to 6 minutes for the tomatoes, and 8 to 10 minutes for the corn. Transfer to a platter as they finish and set aside.
Grill the steak, flipping once and brushing with any leftover mustard mixture, until medium rare (an instant-read thermometer inserted in the thickest part of the steak should read 130°F to 135°F), 12 to 15 minutes. Transfer the steak to a plate, tent with foil, and let rest for 10 minutes.
Meanwhile, slide the skins off the tomatoes. Chop the skins and put them in a small bowl. Add the butter, the remaining 1/2 tsp. mustard, 1/2 tsp. salt, and 1/8 tsp. pepper; mix with a spoon until almost smooth.
Seed and chop the tomatoes; transfer to a large bowl. Cut the corn kernels from the cobs and add to the bowl with the tomatoes. Working over the bowl, scrape the dull side of a table knife down the length of each cob to remove as much of the corn “milk” as possible; discard the cobs. Roughly chop the green beans and scallions and add to the bowl along with the parsley, lime juice, and cream. Toss well and season to taste with salt and pepper.
Thinly slice the steak against the grain, transfer to plates, and brush with some of the tomato butter. Serve with the succotash and remaining tomato butter on the side.