Cut 2″ of the cucumber into ¼” pieces and set aside for serving; coarsely chop remaining cucumber and place in a large bowl. Cut one quarter of the bell pepper into ¼” pieces and set aside for serving; coarsely chop remaining bell pepper and add to bowl with chopped cucumber. Add tomatoes to bowl and toss with shallot, garlic, 2 Tbsp. vinegar, and 1 Tbsp. salt. Let stand 30 minutes at room temperature to let flavors meld.

Transfer tomato mixture along with any accumulated juices to a blender. Add 3 Tbsp. oil and purée on medium speed until smooth; season with salt and vinegar, if desired. Strain through a coarse-mesh sieve into a large bowl or pitcher and chill until cool, about 1 hour.

Divide gazpacho among bowls. Top with cherry tomatoes, chives, reserved cucumber and red pepper pieces, and tomato seeds; drizzle with oil and season with salt. Serve with bread alongside.