This experiment aimed to investigate the possibility to increase the carcass weight of dairy breed lambs and produce moderate-fat meat by applying inexpensive feeding strategies based on restriction and through the use of a fibrous byproduct such as the durum wheat bran (DWB). Sixty-five 45-day-old lambs of the Valle del Belice breed, divided into 6 groups, were fed alfalfa hay supplemented with concentrate feeds including DWB at 0% or 20% (DWB0, DWB20), supplied ad libitum (L) or restricted at 75% (R), and slaughtered at 90 or 120 days of age. The groups were as follows: DWB0-90L (n = 14), DWB20-90L (n = 14), DWB0-120R (n = 10), DWB20-120R (n = 9), DWB0-120L (n = 9), DWB20-120L (n = 9). The diet did not affect feed intake, growth or carcass weight of lambs fed ad libitum, whereas 120-day-old lambs fed DWB associated to restriction showed the lowest weight gain (105 vs. 170, 185 and 190 g/day in DWD20-120R, DWB0-120R, DWB0-120L and DWB20-120L; p = 0.04). The incidence of fat tissue in the hind leg increased (p < 0.0001) from 90L (5.82 and 5.45% with DWB0 and DWB20) to 120R (8.80 and 8.43% with DWB0 and DWB20) and 120L lambs (10.7 and 11.8% with DWB0 and DWB20). Older lambs’ meat, compared to that of 90L lambs, showed analogous levels of intramuscular fat, higher water retention, tenderness and lightness, and a more intense red colour. In meat from 120-day-old lambs, DWB intake tended to reduce the fat level (p = 0.009) and increased polyphenol content (1.10 vs. 1.62, and 1.02 vs. 1.65 g GAE/kg dry matter (DM) in 120R and 120L lambs; p = 0.02), antioxidant capacity (12.8 vs. 14.9, and 12.8 vs. 15.7 mmol trolox eq/kg DM in 120R and 120L lambs; p = 0.02), and the presence of n-3 polyunsaturated fatty acids (FA) (1.61 vs. 2.81, and 1.43 vs. 2.61 g/100 g FA in 120R and 120L lambs; p = 0.007), thereby improving the meat’s health properties. The panelists perceived the effects of DWB inclusion as well as the feeding level with triangle tests.